| L(s) = 1 | + 12·3-s + 9·4-s + 90·9-s + 108·12-s − 12·13-s + 17·16-s + 96·17-s − 288·23-s + 240·25-s + 540·27-s − 384·29-s + 810·36-s − 144·39-s − 1.28e3·43-s + 204·48-s + 592·49-s + 1.15e3·51-s − 108·52-s + 984·53-s + 288·61-s + 153·64-s + 864·68-s − 3.45e3·69-s + 2.88e3·75-s + 4.32e3·79-s + 2.83e3·81-s − 4.60e3·87-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 9/8·4-s + 10/3·9-s + 2.59·12-s − 0.256·13-s + 0.265·16-s + 1.36·17-s − 2.61·23-s + 1.91·25-s + 3.84·27-s − 2.45·29-s + 15/4·36-s − 0.591·39-s − 4.56·43-s + 0.613·48-s + 1.72·49-s + 3.16·51-s − 0.288·52-s + 2.55·53-s + 0.604·61-s + 0.298·64-s + 1.54·68-s − 6.02·69-s + 4.43·75-s + 6.15·79-s + 35/9·81-s − 5.67·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.812172732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.812172732\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p T )^{4} \) |
| 13 | $D_{4}$ | \( 1 + 12 T - 74 p T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 - 9 T^{2} + p^{6} T^{4} - 9 p^{6} T^{6} + p^{12} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 - 48 p T^{2} + 44302 T^{4} - 48 p^{7} T^{6} + p^{12} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 - 592 T^{2} + 310782 T^{4} - 592 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 2172 T^{2} + 3811270 T^{4} - 2172 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 48 T - 1730 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 13552 T^{2} + 94862766 T^{4} - 13552 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + 72 T + p^{3} T^{2} )^{4} \) |
| 29 | $D_{4}$ | \( ( 1 + 192 T + 45862 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 18592 T^{2} + 1722523710 T^{4} - 18592 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 92356 T^{2} + 6285341910 T^{4} - 92356 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 142320 T^{2} + 14548520830 T^{4} - 142320 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 644 T + 250566 T^{2} + 644 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 385164 T^{2} + 58535996374 T^{4} - 385164 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 492 T + 164158 T^{2} - 492 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 758940 T^{2} + 227837028550 T^{4} - 758940 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 144 T + 393094 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 611104 T^{2} + 187596510414 T^{4} - 611104 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 514188 T^{2} + 171350572726 T^{4} - 514188 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 659668 T^{2} + 389934034566 T^{4} - 659668 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 2160 T + 2130910 T^{2} - 2160 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 823068 T^{2} + 604381635622 T^{4} - 823068 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 2258064 T^{2} + 2268670823038 T^{4} - 2258064 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 3513652 T^{2} + 4748412207846 T^{4} - 3513652 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−11.97843303298816643056632639017, −11.17801342567541892636590597802, −11.14759597093676363984320262255, −10.47200883765903349932721561952, −10.29394733648041933706688208559, −9.999201359348123166662267019749, −9.670130327916787056147022595314, −9.271735078580329713532904555382, −9.073746586965703148575828022714, −8.393218968189281304341396093993, −8.223636390196996803376716728956, −8.058888026448256052957670677929, −7.60769323165356513521241440575, −7.18872780027925338184984574409, −6.81759044521424932484784760918, −6.66188265347249156107358484717, −5.96935457729125878076055408982, −5.19839155320078868822551234660, −5.14729400854818857779589962431, −3.96217904350876123059012326389, −3.73023945591655165223810402365, −3.41298872181280410099677272703, −2.49482206629706922676557397874, −2.24565420657345228525944196038, −1.52752033709582105481295709391,
1.52752033709582105481295709391, 2.24565420657345228525944196038, 2.49482206629706922676557397874, 3.41298872181280410099677272703, 3.73023945591655165223810402365, 3.96217904350876123059012326389, 5.14729400854818857779589962431, 5.19839155320078868822551234660, 5.96935457729125878076055408982, 6.66188265347249156107358484717, 6.81759044521424932484784760918, 7.18872780027925338184984574409, 7.60769323165356513521241440575, 8.058888026448256052957670677929, 8.223636390196996803376716728956, 8.393218968189281304341396093993, 9.073746586965703148575828022714, 9.271735078580329713532904555382, 9.670130327916787056147022595314, 9.999201359348123166662267019749, 10.29394733648041933706688208559, 10.47200883765903349932721561952, 11.14759597093676363984320262255, 11.17801342567541892636590597802, 11.97843303298816643056632639017
