| L(s) = 1 | − 14·3-s − 42·5-s − 686·7-s − 698·9-s + 7.42e3·11-s − 1.18e4·13-s + 588·15-s + 1.57e4·17-s + 2.66e4·19-s + 9.60e3·21-s − 3.26e4·23-s − 1.23e5·25-s − 8.33e3·27-s + 1.58e5·29-s + 1.80e5·31-s − 1.03e5·33-s + 2.88e4·35-s + 4.58e4·37-s + 1.65e5·39-s − 3.21e5·41-s + 1.02e6·43-s + 2.93e4·45-s − 1.66e6·47-s + 3.52e5·49-s − 2.21e5·51-s + 4.10e5·53-s − 3.11e5·55-s + ⋯ |
| L(s) = 1 | − 0.299·3-s − 0.150·5-s − 0.755·7-s − 0.319·9-s + 1.68·11-s − 1.49·13-s + 0.0449·15-s + 0.779·17-s + 0.890·19-s + 0.226·21-s − 0.559·23-s − 1.57·25-s − 0.0814·27-s + 1.20·29-s + 1.08·31-s − 0.503·33-s + 0.113·35-s + 0.148·37-s + 0.447·39-s − 0.729·41-s + 1.96·43-s + 0.0479·45-s − 2.34·47-s + 3/7·49-s − 0.233·51-s + 0.378·53-s − 0.252·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 14 T + 298 p T^{2} + 14 p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 42 T + 24986 p T^{2} + 42 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 7428 T + 46542982 T^{2} - 7428 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 70 p^{2} T + 160198410 T^{2} + 70 p^{9} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 15792 T + 526157566 T^{2} - 15792 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 26614 T + 1962247086 T^{2} - 26614 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 32640 T - 991808882 T^{2} + 32640 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 158016 T + 39988772806 T^{2} - 158016 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 180740 T + 38484320958 T^{2} - 180740 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 45824 T - 18085535274 T^{2} - 45824 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 321720 T + 181439440606 T^{2} + 321720 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 1023868 T + 671194246566 T^{2} - 1023868 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1665972 T + 1675477834078 T^{2} + 1665972 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 410628 T + 2334205080574 T^{2} - 410628 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1702134 T + 4026188129518 T^{2} - 1702134 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 547526 T + 5609323300002 T^{2} - 547526 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2590616 T + 4654840470246 T^{2} + 2590616 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4129272 T + 22218672158062 T^{2} + 4129272 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8008868 T + 520866105078 p T^{2} + 8008868 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2470456 T - 13654752819234 T^{2} + 2470456 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 9900786 T + 68835957963214 T^{2} + 9900786 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 15423492 T + 143317325773078 T^{2} - 15423492 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 17377472 T + 164552259333822 T^{2} + 17377472 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−9.814741519227456466095906845817, −9.458288251159763704496608952296, −8.744281578355007269732379765283, −8.507616993048706384383949762351, −7.74365054419240585133325040303, −7.33739388281777716175404789516, −7.05474266829147310803440279533, −6.32244633746117330923072466996, −5.99520558263954939268585437930, −5.67742560633799263990497592285, −4.83288437665211969781915381707, −4.46490010557784388982338280340, −3.88594279432057509837385114541, −3.32804606361564183931641959576, −2.82564342249912263947841432541, −2.24558125878557062632114301329, −1.35397653416552402021279311930, −1.04968408307045502605718800923, 0, 0,
1.04968408307045502605718800923, 1.35397653416552402021279311930, 2.24558125878557062632114301329, 2.82564342249912263947841432541, 3.32804606361564183931641959576, 3.88594279432057509837385114541, 4.46490010557784388982338280340, 4.83288437665211969781915381707, 5.67742560633799263990497592285, 5.99520558263954939268585437930, 6.32244633746117330923072466996, 7.05474266829147310803440279533, 7.33739388281777716175404789516, 7.74365054419240585133325040303, 8.507616993048706384383949762351, 8.744281578355007269732379765283, 9.458288251159763704496608952296, 9.814741519227456466095906845817
