| L(s) = 1 | − 2-s + 1.93·3-s + 4-s + 3.86·5-s − 1.93·6-s − 8-s + 0.732·9-s − 3.86·10-s + 3.46·11-s + 1.93·12-s + 3.34·13-s + 7.46·15-s + 16-s − 0.896·17-s − 0.732·18-s + 0.517·19-s + 3.86·20-s − 3.46·22-s + 3·23-s − 1.93·24-s + 9.92·25-s − 3.34·26-s − 4.38·27-s − 1.73·29-s − 7.46·30-s + 4.52·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.11·3-s + 0.5·4-s + 1.72·5-s − 0.788·6-s − 0.353·8-s + 0.244·9-s − 1.22·10-s + 1.04·11-s + 0.557·12-s + 0.928·13-s + 1.92·15-s + 0.250·16-s − 0.217·17-s − 0.172·18-s + 0.118·19-s + 0.863·20-s − 0.738·22-s + 0.625·23-s − 0.394·24-s + 1.98·25-s − 0.656·26-s − 0.843·27-s − 0.321·29-s − 1.36·30-s + 0.811·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5194 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5194 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.448148749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.448148749\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 - 1.93T + 3T^{2} \) |
| 5 | \( 1 - 3.86T + 5T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 3.34T + 13T^{2} \) |
| 17 | \( 1 + 0.896T + 17T^{2} \) |
| 19 | \( 1 - 0.517T + 19T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + 1.73T + 29T^{2} \) |
| 31 | \( 1 - 4.52T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 - 1.03T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 59 | \( 1 + 7.45T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 7.46T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 + 7.72T + 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 - 7.58T + 83T^{2} \) |
| 89 | \( 1 + 1.41T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
| 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
−8.610474835202164489992321487539, −7.59433721250618784298344714908, −6.82443070452756220250984752147, −6.11139137316727486035401655706, −5.64264485196126254449651653201, −4.41652596499542678865458549803, −3.35544939219804741017815330261, −2.66836414764223535829891630067, −1.82221804342692927391731957747, −1.18836116747609372865694769959,
1.18836116747609372865694769959, 1.82221804342692927391731957747, 2.66836414764223535829891630067, 3.35544939219804741017815330261, 4.41652596499542678865458549803, 5.64264485196126254449651653201, 6.11139137316727486035401655706, 6.82443070452756220250984752147, 7.59433721250618784298344714908, 8.610474835202164489992321487539
