L(s) = 1 | + 2.30·2-s + 1.94·3-s + 3.32·4-s + 4.48·6-s + 0.292·7-s + 3.07·8-s + 0.779·9-s + 5.73·11-s + 6.47·12-s + 6.40·13-s + 0.676·14-s + 0.428·16-s − 1.19·17-s + 1.79·18-s − 6.61·19-s + 0.569·21-s + 13.2·22-s + 4.21·23-s + 5.96·24-s + 14.7·26-s − 4.31·27-s + 0.975·28-s + 7.24·29-s + 5.26·31-s − 5.15·32-s + 11.1·33-s − 2.76·34-s + ⋯ |
L(s) = 1 | + 1.63·2-s + 1.12·3-s + 1.66·4-s + 1.83·6-s + 0.110·7-s + 1.08·8-s + 0.259·9-s + 1.73·11-s + 1.86·12-s + 1.77·13-s + 0.180·14-s + 0.107·16-s − 0.290·17-s + 0.424·18-s − 1.51·19-s + 0.124·21-s + 2.82·22-s + 0.878·23-s + 1.21·24-s + 2.90·26-s − 0.830·27-s + 0.184·28-s + 1.34·29-s + 0.945·31-s − 0.910·32-s + 1.94·33-s − 0.474·34-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
\(L(1)\) |
\(\approx\) |
\(9.120160254\) |
\(L(\frac12)\) |
\(\approx\) |
\(9.120160254\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.30T + 2T^{2} \) |
| 3 | \( 1 - 1.94T + 3T^{2} \) |
| 7 | \( 1 - 0.292T + 7T^{2} \) |
| 11 | \( 1 - 5.73T + 11T^{2} \) |
| 13 | \( 1 - 6.40T + 13T^{2} \) |
| 17 | \( 1 + 1.19T + 17T^{2} \) |
| 19 | \( 1 + 6.61T + 19T^{2} \) |
| 23 | \( 1 - 4.21T + 23T^{2} \) |
| 29 | \( 1 - 7.24T + 29T^{2} \) |
| 31 | \( 1 - 5.26T + 31T^{2} \) |
| 37 | \( 1 + 5.15T + 37T^{2} \) |
| 41 | \( 1 + 4.48T + 41T^{2} \) |
| 43 | \( 1 + 1.68T + 43T^{2} \) |
| 47 | \( 1 + 6.59T + 47T^{2} \) |
| 53 | \( 1 + 0.965T + 53T^{2} \) |
| 59 | \( 1 - 3.89T + 59T^{2} \) |
| 61 | \( 1 + 6.97T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 - 6.70T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 5.98T + 83T^{2} \) |
| 89 | \( 1 - 6.24T + 89T^{2} \) |
| 97 | \( 1 + 4.42T + 97T^{2} \) |
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
−8.318197788030742307770893241622, −6.99630753959386213660554548918, −6.36572436031487449845659220724, −6.18239595185631658309165079297, −4.90561435934138451503832472049, −4.29001158293717572481715215639, −3.57112883793255097651236746407, −3.23353878751479109614084157701, −2.17894969108398726555352934483, −1.35966289820266355507154387482,
1.35966289820266355507154387482, 2.17894969108398726555352934483, 3.23353878751479109614084157701, 3.57112883793255097651236746407, 4.29001158293717572481715215639, 4.90561435934138451503832472049, 6.18239595185631658309165079297, 6.36572436031487449845659220724, 6.99630753959386213660554548918, 8.318197788030742307770893241622