| L(s) = 1 | + 16·2-s + 192·4-s − 250·5-s + 670·7-s + 2.04e3·8-s − 4.00e3·10-s + 570·11-s − 1.18e4·13-s + 1.07e4·14-s + 2.04e4·16-s − 5.16e3·17-s + 1.86e4·19-s − 4.80e4·20-s + 9.12e3·22-s − 9.67e4·23-s + 4.68e4·25-s − 1.89e5·26-s + 1.28e5·28-s − 9.93e4·29-s − 1.50e5·31-s + 1.96e5·32-s − 8.25e4·34-s − 1.67e5·35-s + 3.94e5·37-s + 2.98e5·38-s − 5.12e5·40-s − 2.41e5·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.738·7-s + 1.41·8-s − 1.26·10-s + 0.129·11-s − 1.49·13-s + 1.04·14-s + 5/4·16-s − 0.254·17-s + 0.624·19-s − 1.34·20-s + 0.182·22-s − 1.65·23-s + 3/5·25-s − 2.11·26-s + 1.10·28-s − 0.756·29-s − 0.909·31-s + 1.06·32-s − 0.360·34-s − 0.660·35-s + 1.28·37-s + 0.883·38-s − 1.26·40-s − 0.547·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72900 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 670 T + 1358442 T^{2} - 670 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 570 T + 12475498 T^{2} - 570 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 11870 T + 112216110 T^{2} + 11870 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5160 T + 252823021 T^{2} + 5160 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 18676 T + 98584263 p T^{2} - 18676 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 96786 T + 7842573343 T^{2} + 96786 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 99390 T + 17592475342 T^{2} + 99390 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 150920 T + 50180471097 T^{2} + 150920 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 394900 T + 223295554110 T^{2} - 394900 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 241650 T + 226375649686 T^{2} + 241650 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 88130 T + 537150675714 T^{2} + 88130 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 61596 T + 191178794830 T^{2} + 61596 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 594696 T + 1679945159653 T^{2} + 594696 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3633450 T + 8259065003674 T^{2} + 3633450 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5602154 T + 14129219989071 T^{2} + 5602154 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 307880 T - 2839761751278 T^{2} + 307880 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1734510 T + 18246163338682 T^{2} - 1734510 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7493330 T + 36072611984838 T^{2} + 7493330 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1304396 T + 29267153060997 T^{2} + 1304396 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 3908838 T + 36829760574415 T^{2} + 3908838 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 3346950 T + 8127447098614 T^{2} + 3346950 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 649760 T + 159873152761602 T^{2} + 649760 p^{7} T^{3} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.69196034752604961547346645634, −10.15331543228884956415740556181, −9.471401385841795841525823470567, −9.164008506041746637641828665535, −8.027912851533385391479401411695, −8.003716900544688999658944964269, −7.38107304282155557214243510690, −7.15934405890681364144534124860, −6.17678446798829538612241439947, −6.01079422675238843718164742736, −5.00464003402768331400974649396, −4.95398995262785637963758090527, −4.18538143078229269153094611259, −3.97164393624951599621558394651, −2.98317120419677995405682407444, −2.79035095425028834060811600901, −1.66403516536268690270679870655, −1.57142038149053958642123619892, 0, 0,
1.57142038149053958642123619892, 1.66403516536268690270679870655, 2.79035095425028834060811600901, 2.98317120419677995405682407444, 3.97164393624951599621558394651, 4.18538143078229269153094611259, 4.95398995262785637963758090527, 5.00464003402768331400974649396, 6.01079422675238843718164742736, 6.17678446798829538612241439947, 7.15934405890681364144534124860, 7.38107304282155557214243510690, 8.003716900544688999658944964269, 8.027912851533385391479401411695, 9.164008506041746637641828665535, 9.471401385841795841525823470567, 10.15331543228884956415740556181, 10.69196034752604961547346645634
