| L(s) = 1 | − 3-s + 3.34·5-s + 3.54·7-s + 9-s − 5.22·11-s + 5.27·13-s − 3.34·15-s + 6.86·17-s + 5.25·19-s − 3.54·21-s + 2.32·23-s + 6.21·25-s − 27-s − 0.922·29-s − 10.9·31-s + 5.22·33-s + 11.8·35-s − 10.2·37-s − 5.27·39-s + 2.58·41-s + 4.38·43-s + 3.34·45-s + 6.30·47-s + 5.57·49-s − 6.86·51-s + 10.7·53-s − 17.4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.49·5-s + 1.34·7-s + 0.333·9-s − 1.57·11-s + 1.46·13-s − 0.864·15-s + 1.66·17-s + 1.20·19-s − 0.773·21-s + 0.485·23-s + 1.24·25-s − 0.192·27-s − 0.171·29-s − 1.97·31-s + 0.908·33-s + 2.00·35-s − 1.67·37-s − 0.844·39-s + 0.403·41-s + 0.668·43-s + 0.499·45-s + 0.919·47-s + 0.796·49-s − 0.961·51-s + 1.47·53-s − 2.35·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.767956101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.767956101\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 3.34T + 5T^{2} \) |
| 7 | \( 1 - 3.54T + 7T^{2} \) |
| 11 | \( 1 + 5.22T + 11T^{2} \) |
| 13 | \( 1 - 5.27T + 13T^{2} \) |
| 17 | \( 1 - 6.86T + 17T^{2} \) |
| 19 | \( 1 - 5.25T + 19T^{2} \) |
| 23 | \( 1 - 2.32T + 23T^{2} \) |
| 29 | \( 1 + 0.922T + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 2.58T + 41T^{2} \) |
| 43 | \( 1 - 4.38T + 43T^{2} \) |
| 47 | \( 1 - 6.30T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + 8.19T + 67T^{2} \) |
| 71 | \( 1 + 3.27T + 71T^{2} \) |
| 73 | \( 1 + 9.70T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 1.83T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
| 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
−8.500847209502543645396269417560, −7.59567257782344396253575385579, −7.11531921298937383693255484286, −5.67669191252084673328372080330, −5.55287336878493864753227062362, −5.22841781999259932236872637628, −3.88366741938985556107640371381, −2.80993317753682318872322136568, −1.72853786336969382718225295129, −1.11279951080320927994646744319,
1.11279951080320927994646744319, 1.72853786336969382718225295129, 2.80993317753682318872322136568, 3.88366741938985556107640371381, 5.22841781999259932236872637628, 5.55287336878493864753227062362, 5.67669191252084673328372080330, 7.11531921298937383693255484286, 7.59567257782344396253575385579, 8.500847209502543645396269417560
