| L(s) = 1 | + 2.58·2-s + 3-s + 4.70·4-s + 1.79·5-s + 2.58·6-s − 2.63·7-s + 7.00·8-s + 9-s + 4.65·10-s + 0.775·11-s + 4.70·12-s + 13-s − 6.82·14-s + 1.79·15-s + 8.73·16-s + 5.13·17-s + 2.58·18-s + 1.39·19-s + 8.46·20-s − 2.63·21-s + 2.00·22-s − 2.40·23-s + 7.00·24-s − 1.76·25-s + 2.58·26-s + 27-s − 12.4·28-s + ⋯ |
| L(s) = 1 | + 1.83·2-s + 0.577·3-s + 2.35·4-s + 0.804·5-s + 1.05·6-s − 0.996·7-s + 2.47·8-s + 0.333·9-s + 1.47·10-s + 0.233·11-s + 1.35·12-s + 0.277·13-s − 1.82·14-s + 0.464·15-s + 2.18·16-s + 1.24·17-s + 0.610·18-s + 0.320·19-s + 1.89·20-s − 0.575·21-s + 0.428·22-s − 0.501·23-s + 1.43·24-s − 0.352·25-s + 0.507·26-s + 0.192·27-s − 2.34·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.515304268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.515304268\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 2.58T + 2T^{2} \) |
| 5 | \( 1 - 1.79T + 5T^{2} \) |
| 7 | \( 1 + 2.63T + 7T^{2} \) |
| 11 | \( 1 - 0.775T + 11T^{2} \) |
| 17 | \( 1 - 5.13T + 17T^{2} \) |
| 19 | \( 1 - 1.39T + 19T^{2} \) |
| 23 | \( 1 + 2.40T + 23T^{2} \) |
| 29 | \( 1 + 0.319T + 29T^{2} \) |
| 31 | \( 1 - 9.47T + 31T^{2} \) |
| 37 | \( 1 + 2.39T + 37T^{2} \) |
| 41 | \( 1 + 8.89T + 41T^{2} \) |
| 43 | \( 1 + 4.21T + 43T^{2} \) |
| 47 | \( 1 + 2.15T + 47T^{2} \) |
| 53 | \( 1 - 7.10T + 53T^{2} \) |
| 59 | \( 1 + 3.59T + 59T^{2} \) |
| 61 | \( 1 + 1.72T + 61T^{2} \) |
| 67 | \( 1 - 6.21T + 67T^{2} \) |
| 71 | \( 1 + 5.04T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 - 0.0506T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 - 0.413T + 89T^{2} \) |
| 97 | \( 1 - 3.07T + 97T^{2} \) |
| 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
−8.263928157753766966538305118607, −7.42053876220770401072061249176, −6.57538560891644413415900621043, −6.15917721756087981938300965244, −5.45473920732476321806663943178, −4.66845118095224992184094200911, −3.63974490554820894570411116431, −3.25349501460258255114283872685, −2.41238524173137534446302165600, −1.45020624901248550674223946525,
1.45020624901248550674223946525, 2.41238524173137534446302165600, 3.25349501460258255114283872685, 3.63974490554820894570411116431, 4.66845118095224992184094200911, 5.45473920732476321806663943178, 6.15917721756087981938300965244, 6.57538560891644413415900621043, 7.42053876220770401072061249176, 8.263928157753766966538305118607
