| L(s) = 1 | + 2.24·2-s − 3-s + 3.05·4-s − 2.24·6-s + 1.38·7-s + 2.37·8-s + 9-s + 5.19·11-s − 3.05·12-s − 2.59·13-s + 3.11·14-s − 0.769·16-s + 3.45·17-s + 2.24·18-s + 1.01·19-s − 1.38·21-s + 11.6·22-s + 2.00·23-s − 2.37·24-s − 5.84·26-s − 27-s + 4.23·28-s + 7.23·29-s − 0.286·31-s − 6.48·32-s − 5.19·33-s + 7.76·34-s + ⋯ |
| L(s) = 1 | + 1.59·2-s − 0.577·3-s + 1.52·4-s − 0.918·6-s + 0.523·7-s + 0.840·8-s + 0.333·9-s + 1.56·11-s − 0.882·12-s − 0.720·13-s + 0.832·14-s − 0.192·16-s + 0.837·17-s + 0.530·18-s + 0.233·19-s − 0.302·21-s + 2.48·22-s + 0.417·23-s − 0.485·24-s − 1.14·26-s − 0.192·27-s + 0.800·28-s + 1.34·29-s − 0.0515·31-s − 1.14·32-s − 0.903·33-s + 1.33·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.609470407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.609470407\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 2.24T + 2T^{2} \) |
| 7 | \( 1 - 1.38T + 7T^{2} \) |
| 11 | \( 1 - 5.19T + 11T^{2} \) |
| 13 | \( 1 + 2.59T + 13T^{2} \) |
| 17 | \( 1 - 3.45T + 17T^{2} \) |
| 19 | \( 1 - 1.01T + 19T^{2} \) |
| 23 | \( 1 - 2.00T + 23T^{2} \) |
| 29 | \( 1 - 7.23T + 29T^{2} \) |
| 31 | \( 1 + 0.286T + 31T^{2} \) |
| 37 | \( 1 + 4.63T + 37T^{2} \) |
| 41 | \( 1 + 0.775T + 41T^{2} \) |
| 43 | \( 1 + 2.06T + 43T^{2} \) |
| 53 | \( 1 + 0.703T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 7.46T + 61T^{2} \) |
| 67 | \( 1 - 3.36T + 67T^{2} \) |
| 71 | \( 1 - 3.94T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 + 6.69T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−8.484072421960250590179570324384, −7.47281917737239710062591389554, −6.66929486215968638844548962916, −6.31659682351732760608596352869, −5.17090950305029205874576142665, −4.99750161799842875707322101877, −3.99675570296308343381305901140, −3.39491433842402969077687278683, −2.23648470242647652953422958066, −1.10732671884557889328403335424,
1.10732671884557889328403335424, 2.23648470242647652953422958066, 3.39491433842402969077687278683, 3.99675570296308343381305901140, 4.99750161799842875707322101877, 5.17090950305029205874576142665, 6.31659682351732760608596352869, 6.66929486215968638844548962916, 7.47281917737239710062591389554, 8.484072421960250590179570324384
