| L(s) = 1 | + 0.732·2-s − 1.46·4-s + 4.73·7-s − 2.53·8-s + 5.73·11-s − 1.46·13-s + 3.46·14-s + 1.07·16-s − 2.73·17-s + 4.46·19-s + 4.19·22-s − 3.46·23-s − 1.07·26-s − 6.92·28-s − 3.19·29-s − 3·31-s + 5.85·32-s − 2·34-s + 2.73·37-s + 3.26·38-s + 7.19·41-s − 0.196·43-s − 8.39·44-s − 2.53·46-s − 8.73·47-s + 15.3·49-s + 2.14·52-s + ⋯ |
| L(s) = 1 | + 0.517·2-s − 0.732·4-s + 1.78·7-s − 0.896·8-s + 1.72·11-s − 0.406·13-s + 0.925·14-s + 0.267·16-s − 0.662·17-s + 1.02·19-s + 0.894·22-s − 0.722·23-s − 0.210·26-s − 1.30·28-s − 0.593·29-s − 0.538·31-s + 1.03·32-s − 0.342·34-s + 0.449·37-s + 0.530·38-s + 1.12·41-s − 0.0299·43-s − 1.26·44-s − 0.373·46-s − 1.27·47-s + 2.19·49-s + 0.297·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.443139126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.443139126\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 0.732T + 2T^{2} \) |
| 7 | \( 1 - 4.73T + 7T^{2} \) |
| 11 | \( 1 - 5.73T + 11T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 + 2.73T + 17T^{2} \) |
| 19 | \( 1 - 4.46T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 + 3.19T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 2.73T + 37T^{2} \) |
| 41 | \( 1 - 7.19T + 41T^{2} \) |
| 43 | \( 1 + 0.196T + 43T^{2} \) |
| 47 | \( 1 + 8.73T + 47T^{2} \) |
| 53 | \( 1 - 6.73T + 53T^{2} \) |
| 59 | \( 1 - 8.26T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + 3.46T + 67T^{2} \) |
| 71 | \( 1 - 3.73T + 71T^{2} \) |
| 73 | \( 1 - 7.66T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 - 2.19T + 83T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 - 9.66T + 97T^{2} \) |
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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
−9.178420613225847899306843651419, −8.377417815955795356086287445853, −7.70708037315806450811749373635, −6.72992373634061862250813707979, −5.71049004740771746746258154533, −5.01455882449624897994248053546, −4.26519752515558774275624372848, −3.68340778322905640421676991768, −2.14719958318733903217247076994, −1.05434842431233943681872318850,
1.05434842431233943681872318850, 2.14719958318733903217247076994, 3.68340778322905640421676991768, 4.26519752515558774275624372848, 5.01455882449624897994248053546, 5.71049004740771746746258154533, 6.72992373634061862250813707979, 7.70708037315806450811749373635, 8.377417815955795356086287445853, 9.178420613225847899306843651419
