| L(s) = 1 | + 3.30·3-s − 5-s − 2.55·7-s + 7.93·9-s − 2.72·11-s + 7.12·13-s − 3.30·15-s + 0.924·17-s + 7.51·19-s − 8.43·21-s − 23-s + 25-s + 16.3·27-s − 2.38·29-s + 0.866·31-s − 9.00·33-s + 2.55·35-s + 0.352·37-s + 23.5·39-s + 4.34·41-s − 7.93·45-s − 13.3·47-s − 0.495·49-s + 3.05·51-s + 3.99·53-s + 2.72·55-s + 24.8·57-s + ⋯ |
| L(s) = 1 | + 1.90·3-s − 0.447·5-s − 0.963·7-s + 2.64·9-s − 0.821·11-s + 1.97·13-s − 0.853·15-s + 0.224·17-s + 1.72·19-s − 1.84·21-s − 0.208·23-s + 0.200·25-s + 3.13·27-s − 0.442·29-s + 0.155·31-s − 1.56·33-s + 0.431·35-s + 0.0580·37-s + 3.77·39-s + 0.677·41-s − 1.18·45-s − 1.94·47-s − 0.0707·49-s + 0.427·51-s + 0.548·53-s + 0.367·55-s + 3.29·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.786071565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.786071565\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 3.30T + 3T^{2} \) |
| 7 | \( 1 + 2.55T + 7T^{2} \) |
| 11 | \( 1 + 2.72T + 11T^{2} \) |
| 13 | \( 1 - 7.12T + 13T^{2} \) |
| 17 | \( 1 - 0.924T + 17T^{2} \) |
| 19 | \( 1 - 7.51T + 19T^{2} \) |
| 29 | \( 1 + 2.38T + 29T^{2} \) |
| 31 | \( 1 - 0.866T + 31T^{2} \) |
| 37 | \( 1 - 0.352T + 37T^{2} \) |
| 41 | \( 1 - 4.34T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 - 3.99T + 53T^{2} \) |
| 59 | \( 1 + 3.84T + 59T^{2} \) |
| 61 | \( 1 + 9.14T + 61T^{2} \) |
| 67 | \( 1 + 3.15T + 67T^{2} \) |
| 71 | \( 1 + 6.07T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 6.35T + 83T^{2} \) |
| 89 | \( 1 + 9.71T + 89T^{2} \) |
| 97 | \( 1 - 8.76T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.807896551322860008838534651220, −9.181268409898647740169010277397, −8.381310241584692499435959626850, −7.78039899087033012236440891210, −7.00007863474549330956034237589, −5.84438893151810673248538426668, −4.32983811528020909020172018173, −3.29091365915199499416758983718, −3.07269763056208165100716382094, −1.43596666248003605840752136528,
1.43596666248003605840752136528, 3.07269763056208165100716382094, 3.29091365915199499416758983718, 4.32983811528020909020172018173, 5.84438893151810673248538426668, 7.00007863474549330956034237589, 7.78039899087033012236440891210, 8.381310241584692499435959626850, 9.181268409898647740169010277397, 9.807896551322860008838534651220
