| L(s) = 1 | − 2.64·2-s + 0.795·3-s + 5.02·4-s − 2.10·6-s − 2.31·7-s − 8.00·8-s − 2.36·9-s − 4.06·11-s + 3.99·12-s − 0.795·13-s + 6.13·14-s + 11.1·16-s − 5.31·17-s + 6.27·18-s − 2.77·19-s − 1.83·21-s + 10.7·22-s − 8.05·23-s − 6.36·24-s + 2.10·26-s − 4.26·27-s − 11.6·28-s + 7.71·29-s + 4.04·31-s − 13.5·32-s − 3.23·33-s + 14.0·34-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 0.459·3-s + 2.51·4-s − 0.860·6-s − 0.874·7-s − 2.83·8-s − 0.789·9-s − 1.22·11-s + 1.15·12-s − 0.220·13-s + 1.63·14-s + 2.79·16-s − 1.28·17-s + 1.47·18-s − 0.636·19-s − 0.401·21-s + 2.29·22-s − 1.68·23-s − 1.29·24-s + 0.413·26-s − 0.821·27-s − 2.19·28-s + 1.43·29-s + 0.726·31-s − 2.40·32-s − 0.562·33-s + 2.41·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1426966851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1426966851\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 197 | \( 1 + T \) |
| good | 2 | \( 1 + 2.64T + 2T^{2} \) |
| 3 | \( 1 - 0.795T + 3T^{2} \) |
| 7 | \( 1 + 2.31T + 7T^{2} \) |
| 11 | \( 1 + 4.06T + 11T^{2} \) |
| 13 | \( 1 + 0.795T + 13T^{2} \) |
| 17 | \( 1 + 5.31T + 17T^{2} \) |
| 19 | \( 1 + 2.77T + 19T^{2} \) |
| 23 | \( 1 + 8.05T + 23T^{2} \) |
| 29 | \( 1 - 7.71T + 29T^{2} \) |
| 31 | \( 1 - 4.04T + 31T^{2} \) |
| 37 | \( 1 - 6.57T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 7.11T + 43T^{2} \) |
| 47 | \( 1 - 2.38T + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 - 9.52T + 61T^{2} \) |
| 67 | \( 1 + 2.42T + 67T^{2} \) |
| 71 | \( 1 + 8.31T + 71T^{2} \) |
| 73 | \( 1 + 16.3T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 2.69T + 83T^{2} \) |
| 89 | \( 1 + 3.57T + 89T^{2} \) |
| 97 | \( 1 + 5.43T + 97T^{2} \) |
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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
−8.440676507145179809697855778806, −7.927888572077619365719329143166, −6.99353759662930450860840721712, −6.40420037495449815775459067689, −5.79760316355393600097993593646, −4.50054266563053600645886641969, −3.12410297869396013528651558164, −2.61050999878008539025472366918, −1.89453924246241008216507570086, −0.24530493089577228690068579332,
0.24530493089577228690068579332, 1.89453924246241008216507570086, 2.61050999878008539025472366918, 3.12410297869396013528651558164, 4.50054266563053600645886641969, 5.79760316355393600097993593646, 6.40420037495449815775459067689, 6.99353759662930450860840721712, 7.927888572077619365719329143166, 8.440676507145179809697855778806
