L(s) = 1 | − 2-s + 3·3-s + 4-s + 5-s − 3·6-s − 8-s + 6·9-s − 10-s − 11-s + 3·12-s − 13-s + 3·15-s + 16-s − 6·17-s − 6·18-s + 4·19-s + 20-s + 22-s + 4·23-s − 3·24-s + 25-s + 26-s + 9·27-s + 29-s − 3·30-s + 8·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s + 0.447·5-s − 1.22·6-s − 0.353·8-s + 2·9-s − 0.316·10-s − 0.301·11-s + 0.866·12-s − 0.277·13-s + 0.774·15-s + 1/4·16-s − 1.45·17-s − 1.41·18-s + 0.917·19-s + 0.223·20-s + 0.213·22-s + 0.834·23-s − 0.612·24-s + 1/5·25-s + 0.196·26-s + 1.73·27-s + 0.185·29-s − 0.547·30-s + 1.43·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.107708427\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.107708427\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.251149346733042942492977589760, −7.74133801652546994485982366154, −6.94276050503055133980493926422, −6.42440455050500010181111455159, −5.15923748814557334977102614763, −4.36705184362758752032716864732, −3.33862737172168888996434617371, −2.62304020585448043093828955884, −2.11228695866909905149924738319, −0.997083390465878268714703350219,
0.997083390465878268714703350219, 2.11228695866909905149924738319, 2.62304020585448043093828955884, 3.33862737172168888996434617371, 4.36705184362758752032716864732, 5.15923748814557334977102614763, 6.42440455050500010181111455159, 6.94276050503055133980493926422, 7.74133801652546994485982366154, 8.251149346733042942492977589760