| L(s) = 1 | − 2-s − 2.05·3-s + 4-s − 2.11·5-s + 2.05·6-s − 1.54·7-s − 8-s + 1.21·9-s + 2.11·10-s + 1.93·11-s − 2.05·12-s − 0.601·13-s + 1.54·14-s + 4.34·15-s + 16-s + 2.46·17-s − 1.21·18-s + 8.02·19-s − 2.11·20-s + 3.16·21-s − 1.93·22-s + 6.61·23-s + 2.05·24-s − 0.513·25-s + 0.601·26-s + 3.66·27-s − 1.54·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.18·3-s + 0.5·4-s − 0.947·5-s + 0.838·6-s − 0.583·7-s − 0.353·8-s + 0.404·9-s + 0.669·10-s + 0.584·11-s − 0.592·12-s − 0.166·13-s + 0.412·14-s + 1.12·15-s + 0.250·16-s + 0.597·17-s − 0.286·18-s + 1.84·19-s − 0.473·20-s + 0.691·21-s − 0.413·22-s + 1.37·23-s + 0.419·24-s − 0.102·25-s + 0.118·26-s + 0.705·27-s − 0.291·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6382566780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6382566780\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 + 2.05T + 3T^{2} \) |
| 5 | \( 1 + 2.11T + 5T^{2} \) |
| 7 | \( 1 + 1.54T + 7T^{2} \) |
| 11 | \( 1 - 1.93T + 11T^{2} \) |
| 13 | \( 1 + 0.601T + 13T^{2} \) |
| 17 | \( 1 - 2.46T + 17T^{2} \) |
| 19 | \( 1 - 8.02T + 19T^{2} \) |
| 23 | \( 1 - 6.61T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 37 | \( 1 - 2.60T + 37T^{2} \) |
| 41 | \( 1 + 6.32T + 41T^{2} \) |
| 43 | \( 1 - 1.64T + 43T^{2} \) |
| 47 | \( 1 + 8.52T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 - 9.73T + 59T^{2} \) |
| 61 | \( 1 + 2.23T + 61T^{2} \) |
| 67 | \( 1 + 7.05T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 2.93T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 1.08T + 89T^{2} \) |
| 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
−8.054733506461792827695439879251, −7.27331024655509214768856922784, −6.74071076647906805344272597538, −6.13212207053012867816422339640, −5.22554051016122767000454803436, −4.67355757029073027583246439114, −3.41725381185608582229526985197, −2.99698143424494114914670666392, −1.28173149055349977788382724001, −0.57001064866695654761553578975,
0.57001064866695654761553578975, 1.28173149055349977788382724001, 2.99698143424494114914670666392, 3.41725381185608582229526985197, 4.67355757029073027583246439114, 5.22554051016122767000454803436, 6.13212207053012867816422339640, 6.74071076647906805344272597538, 7.27331024655509214768856922784, 8.054733506461792827695439879251
