| L(s) = 1 | + 1.98·2-s − 2.93·3-s + 1.94·4-s − 5.82·6-s + 3.87·7-s − 0.108·8-s + 5.60·9-s + 5.43·11-s − 5.70·12-s − 4.51·13-s + 7.70·14-s − 4.10·16-s + 3.56·17-s + 11.1·18-s − 4.95·19-s − 11.3·21-s + 10.7·22-s + 8.91·23-s + 0.319·24-s − 8.96·26-s − 7.64·27-s + 7.54·28-s + 2.56·29-s − 1.87·31-s − 7.93·32-s − 15.9·33-s + 7.08·34-s + ⋯ |
| L(s) = 1 | + 1.40·2-s − 1.69·3-s + 0.972·4-s − 2.37·6-s + 1.46·7-s − 0.0385·8-s + 1.86·9-s + 1.63·11-s − 1.64·12-s − 1.25·13-s + 2.05·14-s − 1.02·16-s + 0.864·17-s + 2.62·18-s − 1.13·19-s − 2.48·21-s + 2.30·22-s + 1.85·23-s + 0.0652·24-s − 1.75·26-s − 1.47·27-s + 1.42·28-s + 0.475·29-s − 0.336·31-s − 1.40·32-s − 2.77·33-s + 1.21·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.021811807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.021811807\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 1.98T + 2T^{2} \) |
| 3 | \( 1 + 2.93T + 3T^{2} \) |
| 7 | \( 1 - 3.87T + 7T^{2} \) |
| 11 | \( 1 - 5.43T + 11T^{2} \) |
| 13 | \( 1 + 4.51T + 13T^{2} \) |
| 17 | \( 1 - 3.56T + 17T^{2} \) |
| 19 | \( 1 + 4.95T + 19T^{2} \) |
| 23 | \( 1 - 8.91T + 23T^{2} \) |
| 29 | \( 1 - 2.56T + 29T^{2} \) |
| 31 | \( 1 + 1.87T + 31T^{2} \) |
| 37 | \( 1 - 8.26T + 37T^{2} \) |
| 41 | \( 1 - 1.60T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 0.538T + 47T^{2} \) |
| 53 | \( 1 + 6.76T + 53T^{2} \) |
| 59 | \( 1 + 7.99T + 59T^{2} \) |
| 61 | \( 1 + 6.30T + 61T^{2} \) |
| 67 | \( 1 + 6.40T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 - 8.98T + 73T^{2} \) |
| 79 | \( 1 + 5.39T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 3.42T + 97T^{2} \) |
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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
−7.61100172555496663991162971674, −7.08645367804496033633698826748, −6.28172410819005120145471078633, −5.84035376301579429977461384063, −5.06103846702579792334731023272, −4.46149665289287517366479822913, −4.35725974688889957896908411311, −3.00182617263520924260826693970, −1.77718353718584589059671495790, −0.853186127886074270079382635896,
0.853186127886074270079382635896, 1.77718353718584589059671495790, 3.00182617263520924260826693970, 4.35725974688889957896908411311, 4.46149665289287517366479822913, 5.06103846702579792334731023272, 5.84035376301579429977461384063, 6.28172410819005120145471078633, 7.08645367804496033633698826748, 7.61100172555496663991162971674
