| L(s) = 1 | − 2·2-s − 2·4-s + 8·8-s + 6·9-s − 32·11-s − 4·16-s + 8·17-s − 12·18-s + 32·19-s + 64·22-s + 8·32-s − 16·34-s − 12·36-s − 64·38-s + 40·41-s − 32·43-s + 64·44-s + 76·49-s − 128·59-s − 24·64-s + 256·67-s − 16·68-s + 48·72-s − 200·73-s − 64·76-s + 27·81-s − 80·82-s + ⋯ |
| L(s) = 1 | − 2-s − 1/2·4-s + 8-s + 2/3·9-s − 2.90·11-s − 1/4·16-s + 8/17·17-s − 2/3·18-s + 1.68·19-s + 2.90·22-s + 1/4·32-s − 0.470·34-s − 1/3·36-s − 1.68·38-s + 0.975·41-s − 0.744·43-s + 1.45·44-s + 1.55·49-s − 2.16·59-s − 3/8·64-s + 3.82·67-s − 0.235·68-s + 2/3·72-s − 2.73·73-s − 0.842·76-s + 1/3·81-s − 0.975·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.238403327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.238403327\) |
| \(L(2)\) |
|
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 | $D_{4}$ | \( 1 + p T + 3 p T^{2} + p^{3} T^{3} + p^{4} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2 \wr C_2$ | \( 1 - 76 T^{2} + 3174 T^{4} - 76 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 292 T^{2} + 75366 T^{4} - 292 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 390 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 16 T + 738 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 1636 T^{2} + 1179654 T^{4} - 1636 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 1756 T^{2} + 1539558 T^{4} - 1756 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 460 T^{2} - 683610 T^{4} - 460 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 4612 T^{2} + 8955366 T^{4} - 4612 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 20 T + 1734 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 16 T + 3330 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 5284 T^{2} + 13593798 T^{4} - 5284 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 9436 T^{2} + 38033574 T^{4} - 9436 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 64 T + 7554 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 11332 T^{2} + 56649510 T^{4} - 11332 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 128 T + 12642 T^{2} - 128 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 11236 T^{2} + 75307398 T^{4} - 11236 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 100 T + 10086 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 17548 T^{2} + 151931046 T^{4} - 17548 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 80 T + 14610 T^{2} + 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 100 T + 11430 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 28 T + 12102 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−7.61542258840329012645970535477, −7.29195110224850485989600912554, −7.28349453185847401813681058526, −6.91399723081180882509068551095, −6.59112947884657261563348688267, −6.32796800363346228714449794889, −5.91748006631100659906437364949, −5.61353949750730808895608274708, −5.39284205091534404044395214331, −5.35646418196184000051858942894, −5.27320611641714452340947286904, −4.58310921317889760398765734495, −4.46664492012255547682013110355, −4.40976640438293000042643650391, −4.07045094647321830864306141037, −3.48509384579519590907906963807, −3.09996136569769432226779285703, −3.08449994843024217978156477494, −2.82264116544843334261125194993, −2.24597278411267552300654334054, −2.05328575705694112575270049135, −1.51763339870187383100952018830, −1.08383281288736366467184932869, −0.50722920726544565022413562557, −0.44990209653661229181022154708,
0.44990209653661229181022154708, 0.50722920726544565022413562557, 1.08383281288736366467184932869, 1.51763339870187383100952018830, 2.05328575705694112575270049135, 2.24597278411267552300654334054, 2.82264116544843334261125194993, 3.08449994843024217978156477494, 3.09996136569769432226779285703, 3.48509384579519590907906963807, 4.07045094647321830864306141037, 4.40976640438293000042643650391, 4.46664492012255547682013110355, 4.58310921317889760398765734495, 5.27320611641714452340947286904, 5.35646418196184000051858942894, 5.39284205091534404044395214331, 5.61353949750730808895608274708, 5.91748006631100659906437364949, 6.32796800363346228714449794889, 6.59112947884657261563348688267, 6.91399723081180882509068551095, 7.28349453185847401813681058526, 7.29195110224850485989600912554, 7.61542258840329012645970535477
