| L(s) = 1 | − 1.24·2-s + 3.09·3-s − 0.453·4-s + 5-s − 3.84·6-s + 3.83·7-s + 3.05·8-s + 6.57·9-s − 1.24·10-s + 11-s − 1.40·12-s − 2.44·13-s − 4.77·14-s + 3.09·15-s − 2.88·16-s + 4.44·17-s − 8.17·18-s + 19-s − 0.453·20-s + 11.8·21-s − 1.24·22-s − 5.34·23-s + 9.44·24-s + 25-s + 3.04·26-s + 11.0·27-s − 1.73·28-s + ⋯ |
| L(s) = 1 | − 0.879·2-s + 1.78·3-s − 0.226·4-s + 0.447·5-s − 1.57·6-s + 1.45·7-s + 1.07·8-s + 2.19·9-s − 0.393·10-s + 0.301·11-s − 0.404·12-s − 0.678·13-s − 1.27·14-s + 0.799·15-s − 0.722·16-s + 1.07·17-s − 1.92·18-s + 0.229·19-s − 0.101·20-s + 2.59·21-s − 0.265·22-s − 1.11·23-s + 1.92·24-s + 0.200·25-s + 0.596·26-s + 2.12·27-s − 0.328·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.208125736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.208125736\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.24T + 2T^{2} \) |
| 3 | \( 1 - 3.09T + 3T^{2} \) |
| 7 | \( 1 - 3.83T + 7T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 - 4.44T + 17T^{2} \) |
| 23 | \( 1 + 5.34T + 23T^{2} \) |
| 29 | \( 1 + 9.42T + 29T^{2} \) |
| 31 | \( 1 + 2.25T + 31T^{2} \) |
| 37 | \( 1 + 9.50T + 37T^{2} \) |
| 41 | \( 1 - 6.98T + 41T^{2} \) |
| 43 | \( 1 - 3.25T + 43T^{2} \) |
| 47 | \( 1 + 4.99T + 47T^{2} \) |
| 53 | \( 1 + 6.72T + 53T^{2} \) |
| 59 | \( 1 - 3.81T + 59T^{2} \) |
| 61 | \( 1 - 7.33T + 61T^{2} \) |
| 67 | \( 1 + 8.95T + 67T^{2} \) |
| 71 | \( 1 - 4.94T + 71T^{2} \) |
| 73 | \( 1 + 3.88T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 - 8.59T + 83T^{2} \) |
| 89 | \( 1 + 3.34T + 89T^{2} \) |
| 97 | \( 1 + 8.42T + 97T^{2} \) |
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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
−9.685274313389910557986527805370, −9.021984887550958660319447306755, −8.361701875090574427782224602217, −7.67415464462137296410156852086, −7.32285155211203615617142284119, −5.46754464008350115464775600572, −4.47446599983597360721742338654, −3.56203332613574628175180388467, −2.09893688182610204554209309753, −1.50435131524585061253398503443,
1.50435131524585061253398503443, 2.09893688182610204554209309753, 3.56203332613574628175180388467, 4.47446599983597360721742338654, 5.46754464008350115464775600572, 7.32285155211203615617142284119, 7.67415464462137296410156852086, 8.361701875090574427782224602217, 9.021984887550958660319447306755, 9.685274313389910557986527805370
