| L(s) = 1 | + 8·2-s + 40·4-s − 8·5-s + 160·8-s − 8·9-s − 64·10-s − 24·11-s + 560·16-s − 64·18-s − 24·19-s − 320·20-s − 192·22-s − 40·23-s + 46·25-s − 40·27-s + 4·29-s + 56·31-s + 1.79e3·32-s − 320·36-s − 40·37-s − 192·38-s − 1.28e3·40-s + 44·41-s + 72·43-s − 960·44-s + 64·45-s − 320·46-s + ⋯ |
| L(s) = 1 | + 4·2-s + 10·4-s − 8/5·5-s + 20·8-s − 8/9·9-s − 6.39·10-s − 2.18·11-s + 35·16-s − 3.55·18-s − 1.26·19-s − 16·20-s − 8.72·22-s − 1.73·23-s + 1.83·25-s − 1.48·27-s + 4/29·29-s + 1.80·31-s + 56·32-s − 8.88·36-s − 1.08·37-s − 5.05·38-s − 32·40-s + 1.07·41-s + 1.67·43-s − 21.8·44-s + 1.42·45-s − 6.95·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21381376 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21381376 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(14.83749604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.83749604\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p T )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 8 T^{2} + 40 T^{3} + 32 T^{4} + 40 p^{2} T^{5} + 8 p^{4} T^{6} + p^{8} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 8 T + 18 T^{2} - 104 T^{3} - 798 T^{4} - 104 p^{2} T^{5} + 18 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 8 T^{2} - 360 T^{3} + 32 T^{4} - 360 p^{2} T^{5} + 8 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 24 T + 216 T^{2} + 864 T^{3} + 2592 T^{4} + 864 p^{2} T^{5} + 216 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 544 T^{2} + 127906 T^{4} - 544 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 792 T^{3} - 105406 T^{4} + 792 p^{2} T^{5} + 288 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 40 T + 408 T^{2} - 23120 T^{3} - 854368 T^{4} - 23120 p^{2} T^{5} + 408 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T + 1686 T^{2} - 356 p T^{3} + 1435682 T^{4} - 356 p^{3} T^{5} + 1686 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 56 T + 1176 T^{2} - 10976 T^{3} + 76832 T^{4} - 10976 p^{2} T^{5} + 1176 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 40 T + 2578 T^{2} + 130040 T^{3} + 5039842 T^{4} + 130040 p^{2} T^{5} + 2578 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 44 T + 646 T^{2} + 108596 T^{3} - 4883198 T^{4} + 108596 p^{2} T^{5} + 646 p^{4} T^{6} - 44 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 72 T + 2592 T^{2} - 150984 T^{3} + 8733314 T^{4} - 150984 p^{2} T^{5} + 2592 p^{4} T^{6} - 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $C_2$ | \( ( 1 - 60 T + p^{2} T^{2} )^{4} \) |
| 53 | $D_4\times C_2$ | \( 1 + 92 T + 4232 T^{2} + 318964 T^{3} + 23607214 T^{4} + 318964 p^{2} T^{5} + 4232 p^{4} T^{6} + 92 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 52 T + 1352 T^{2} + 52 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 104 T + 4626 T^{2} - 187384 T^{3} + 9399522 T^{4} - 187384 p^{2} T^{5} + 4626 p^{4} T^{6} - 104 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 7676 T^{2} + 54213286 T^{4} + 7676 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 304 T + 45576 T^{2} + 4605784 T^{3} + 361971872 T^{4} + 4605784 p^{2} T^{5} + 45576 p^{4} T^{6} + 304 p^{6} T^{7} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 60 T + 1478 T^{2} - 562500 T^{3} - 34738558 T^{4} - 562500 p^{2} T^{5} + 1478 p^{4} T^{6} + 60 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 264 T + 39896 T^{2} + 4405824 T^{3} + 392774432 T^{4} + 4405824 p^{2} T^{5} + 39896 p^{4} T^{6} + 264 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 152 T + 11552 T^{2} + 513304 T^{3} + 10474114 T^{4} + 513304 p^{2} T^{5} + 11552 p^{4} T^{6} + 152 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 1392 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 160 T + 21538 T^{2} - 2865440 T^{3} + 302880322 T^{4} - 2865440 p^{2} T^{5} + 21538 p^{4} T^{6} - 160 p^{6} T^{7} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.14773867554974143217907511017, −10.62276294187313185877052997040, −10.49153313204400739004398426752, −10.22714242869780367194456284910, −10.15227516652542306402585065820, −9.164092654531113451333282390889, −8.527716611574130547311668400605, −8.160852812552420341082080528964, −8.066249291661477140186207698894, −7.49791474040058259221730378111, −7.41577286465334660984286089017, −7.23080681697822765184404775687, −6.76407099646839169495928595395, −6.05446536232189823550786237544, −5.80121809242508897286773588425, −5.67353703863993699671871709338, −5.62646647872910393501643984042, −4.71850785075218066040338711338, −4.30262382388915473783510511805, −4.21034214605787959375093268017, −4.16593436935830806188847977146, −3.18200742452622532072050588063, −2.79900597783308019502898664980, −2.71293274601877252000036473065, −1.97221676032077694254899098108,
1.97221676032077694254899098108, 2.71293274601877252000036473065, 2.79900597783308019502898664980, 3.18200742452622532072050588063, 4.16593436935830806188847977146, 4.21034214605787959375093268017, 4.30262382388915473783510511805, 4.71850785075218066040338711338, 5.62646647872910393501643984042, 5.67353703863993699671871709338, 5.80121809242508897286773588425, 6.05446536232189823550786237544, 6.76407099646839169495928595395, 7.23080681697822765184404775687, 7.41577286465334660984286089017, 7.49791474040058259221730378111, 8.066249291661477140186207698894, 8.160852812552420341082080528964, 8.527716611574130547311668400605, 9.164092654531113451333282390889, 10.15227516652542306402585065820, 10.22714242869780367194456284910, 10.49153313204400739004398426752, 10.62276294187313185877052997040, 11.14773867554974143217907511017
