| L(s) = 1 | − 82·5-s + 456·7-s + 2.52e3·11-s + 1.07e4·13-s + 1.11e4·17-s + 4.12e3·19-s + 8.17e4·23-s − 7.14e4·25-s + 9.97e4·29-s + 4.04e4·31-s − 3.73e4·35-s + 4.19e5·37-s − 1.41e5·41-s − 6.90e5·43-s − 6.82e5·47-s − 6.15e5·49-s + 1.81e6·53-s − 2.06e5·55-s + 9.66e5·59-s − 1.88e6·61-s − 8.83e5·65-s + 2.96e6·67-s − 2.54e6·71-s − 1.68e6·73-s + 1.15e6·77-s − 4.03e6·79-s + 5.38e6·83-s + ⋯ |
| L(s) = 1 | − 0.293·5-s + 0.502·7-s + 0.571·11-s + 1.36·13-s + 0.550·17-s + 0.137·19-s + 1.40·23-s − 0.913·25-s + 0.759·29-s + 0.244·31-s − 0.147·35-s + 1.36·37-s − 0.320·41-s − 1.32·43-s − 0.958·47-s − 0.747·49-s + 1.67·53-s − 0.167·55-s + 0.612·59-s − 1.06·61-s − 0.399·65-s + 1.20·67-s − 0.844·71-s − 0.505·73-s + 0.287·77-s − 0.921·79-s + 1.03·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.917943943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.917943943\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 82 T + p^{7} T^{2} \) |
| 7 | \( 1 - 456 T + p^{7} T^{2} \) |
| 11 | \( 1 - 2524 T + p^{7} T^{2} \) |
| 13 | \( 1 - 10778 T + p^{7} T^{2} \) |
| 17 | \( 1 - 11150 T + p^{7} T^{2} \) |
| 19 | \( 1 - 4124 T + p^{7} T^{2} \) |
| 23 | \( 1 - 81704 T + p^{7} T^{2} \) |
| 29 | \( 1 - 99798 T + p^{7} T^{2} \) |
| 31 | \( 1 - 40480 T + p^{7} T^{2} \) |
| 37 | \( 1 - 419442 T + p^{7} T^{2} \) |
| 41 | \( 1 + 141402 T + p^{7} T^{2} \) |
| 43 | \( 1 + 690428 T + p^{7} T^{2} \) |
| 47 | \( 1 + 682032 T + p^{7} T^{2} \) |
| 53 | \( 1 - 1813118 T + p^{7} T^{2} \) |
| 59 | \( 1 - 966028 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1887670 T + p^{7} T^{2} \) |
| 67 | \( 1 - 2965868 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2548232 T + p^{7} T^{2} \) |
| 73 | \( 1 + 1680326 T + p^{7} T^{2} \) |
| 79 | \( 1 + 4038064 T + p^{7} T^{2} \) |
| 83 | \( 1 - 5385764 T + p^{7} T^{2} \) |
| 89 | \( 1 - 6473046 T + p^{7} T^{2} \) |
| 97 | \( 1 + 6065758 T + p^{7} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.552975603232897470848418589867, −8.598526845692192510880991757679, −7.980260997738736541829598682213, −6.88808253992869487355510280827, −6.02390550697264592778402047490, −4.96057308224437440249545068621, −3.95044708842449749935144025533, −3.05521636261578188440299484107, −1.58975437713899942455776316206, −0.794075146025589823544590102017,
0.794075146025589823544590102017, 1.58975437713899942455776316206, 3.05521636261578188440299484107, 3.95044708842449749935144025533, 4.96057308224437440249545068621, 6.02390550697264592778402047490, 6.88808253992869487355510280827, 7.980260997738736541829598682213, 8.598526845692192510880991757679, 9.552975603232897470848418589867
