| L(s) = 1 | + 500·5-s − 729·9-s − 7.98e3·11-s + 5.06e4·19-s + 1.71e5·25-s + 3.05e5·29-s − 2.47e5·31-s + 7.93e5·41-s − 3.64e5·45-s + 1.12e6·49-s − 3.99e6·55-s + 6.04e5·59-s − 5.66e6·61-s − 2.01e6·71-s − 1.50e7·79-s + 5.31e5·81-s + 1.53e7·89-s + 2.53e7·95-s + 5.82e6·99-s + 2.25e7·101-s − 2.34e7·109-s + 8.88e6·121-s + 4.68e7·125-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1/3·9-s − 1.80·11-s + 1.69·19-s + 11/5·25-s + 2.32·29-s − 1.49·31-s + 1.79·41-s − 0.596·45-s + 1.36·49-s − 3.23·55-s + 0.383·59-s − 3.19·61-s − 0.668·71-s − 3.43·79-s + 1/9·81-s + 2.30·89-s + 3.02·95-s + 0.603·99-s + 2.18·101-s − 1.73·109-s + 0.455·121-s + 2.14·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.336372814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.336372814\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{6} T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 p^{3} T + p^{7} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 1125802 T^{2} + p^{14} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3994 T + p^{7} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 116316134 T^{2} + p^{14} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 397058622 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 25320 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2367161790 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 152664 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 123776 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 75696805270 T^{2} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 396530 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 347519328310 T^{2} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 984199174302 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 813245470198 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 302354 T + p^{7} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2830198 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 1875692967338 T^{2} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 1007580 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 16312522745698 T^{2} + p^{14} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 7517832 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 26186045040870 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 7650250 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 60474559225090 T^{2} + p^{14} T^{4} \) |
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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
−13.69907881933056905380746212461, −13.57094730272544282526456505747, −12.83098248408564801401496230937, −12.41825149665973258205318738399, −11.62104191358790046833974538432, −10.78657078233898953820204984010, −10.33701289467929913579551276087, −10.02847936399806316095204157874, −9.135646224620786249404200720770, −8.884750044664893537439808047900, −7.76991128415772676410459069650, −7.38712905254115588818945735795, −6.34841656340094580033470517336, −5.68894304045933600996659773044, −5.33618482862279306802463284775, −4.57398884338298305398794221574, −2.92786830009403208169906627432, −2.72051907067689143569367878956, −1.64769298444566613509739289572, −0.68376097396052907161815049635,
0.68376097396052907161815049635, 1.64769298444566613509739289572, 2.72051907067689143569367878956, 2.92786830009403208169906627432, 4.57398884338298305398794221574, 5.33618482862279306802463284775, 5.68894304045933600996659773044, 6.34841656340094580033470517336, 7.38712905254115588818945735795, 7.76991128415772676410459069650, 8.884750044664893537439808047900, 9.135646224620786249404200720770, 10.02847936399806316095204157874, 10.33701289467929913579551276087, 10.78657078233898953820204984010, 11.62104191358790046833974538432, 12.41825149665973258205318738399, 12.83098248408564801401496230937, 13.57094730272544282526456505747, 13.69907881933056905380746212461
