| L(s) = 1 | − 2-s + 0.782·3-s + 4-s − 1.76·5-s − 0.782·6-s + 2.34·7-s − 8-s − 2.38·9-s + 1.76·10-s + 5.03·11-s + 0.782·12-s − 2.34·14-s − 1.38·15-s + 16-s + 5.98·17-s + 2.38·18-s − 19-s − 1.76·20-s + 1.83·21-s − 5.03·22-s + 2.35·23-s − 0.782·24-s − 1.88·25-s − 4.21·27-s + 2.34·28-s − 4.79·29-s + 1.38·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.451·3-s + 0.5·4-s − 0.789·5-s − 0.319·6-s + 0.884·7-s − 0.353·8-s − 0.795·9-s + 0.557·10-s + 1.51·11-s + 0.225·12-s − 0.625·14-s − 0.356·15-s + 0.250·16-s + 1.45·17-s + 0.562·18-s − 0.229·19-s − 0.394·20-s + 0.399·21-s − 1.07·22-s + 0.491·23-s − 0.159·24-s − 0.377·25-s − 0.811·27-s + 0.442·28-s − 0.890·29-s + 0.252·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.710148023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.710148023\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.782T + 3T^{2} \) |
| 5 | \( 1 + 1.76T + 5T^{2} \) |
| 7 | \( 1 - 2.34T + 7T^{2} \) |
| 11 | \( 1 - 5.03T + 11T^{2} \) |
| 17 | \( 1 - 5.98T + 17T^{2} \) |
| 23 | \( 1 - 2.35T + 23T^{2} \) |
| 29 | \( 1 + 4.79T + 29T^{2} \) |
| 31 | \( 1 - 0.870T + 31T^{2} \) |
| 37 | \( 1 - 6.18T + 37T^{2} \) |
| 41 | \( 1 - 5.23T + 41T^{2} \) |
| 43 | \( 1 - 0.868T + 43T^{2} \) |
| 47 | \( 1 - 0.145T + 47T^{2} \) |
| 53 | \( 1 - 4.67T + 53T^{2} \) |
| 59 | \( 1 - 3.48T + 59T^{2} \) |
| 61 | \( 1 + 1.64T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 5.29T + 71T^{2} \) |
| 73 | \( 1 - 1.44T + 73T^{2} \) |
| 79 | \( 1 - 6.95T + 79T^{2} \) |
| 83 | \( 1 + 2.22T + 83T^{2} \) |
| 89 | \( 1 - 0.111T + 89T^{2} \) |
| 97 | \( 1 + 8.21T + 97T^{2} \) |
| 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
−8.108660964410869798037482219570, −7.60283763471829872861695002110, −6.84893189680161396765007063872, −5.95993361203771680731204419865, −5.27945335515209894254824889490, −4.11808692862238655741348572265, −3.64104856794029023719866840738, −2.69583435079971994702690764107, −1.66274337717372438135750900397, −0.77537666972459292751234395026,
0.77537666972459292751234395026, 1.66274337717372438135750900397, 2.69583435079971994702690764107, 3.64104856794029023719866840738, 4.11808692862238655741348572265, 5.27945335515209894254824889490, 5.95993361203771680731204419865, 6.84893189680161396765007063872, 7.60283763471829872861695002110, 8.108660964410869798037482219570
