| L(s) = 1 | − 2.30·2-s + 3.32·4-s + 2.49·5-s + 1.60·7-s − 3.05·8-s − 5.76·10-s − 0.733·11-s − 1.89·13-s − 3.69·14-s + 0.404·16-s + 4.37·17-s − 4.63·19-s + 8.30·20-s + 1.69·22-s + 7.11·23-s + 1.24·25-s + 4.37·26-s + 5.32·28-s + 0.128·29-s + 5.18·32-s − 10.1·34-s + 4.00·35-s + 8.42·37-s + 10.7·38-s − 7.63·40-s + 7.37·41-s + 0.230·43-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 1.66·4-s + 1.11·5-s + 0.605·7-s − 1.08·8-s − 1.82·10-s − 0.221·11-s − 0.525·13-s − 0.987·14-s + 0.101·16-s + 1.06·17-s − 1.06·19-s + 1.85·20-s + 0.360·22-s + 1.48·23-s + 0.248·25-s + 0.858·26-s + 1.00·28-s + 0.0239·29-s + 0.915·32-s − 1.73·34-s + 0.676·35-s + 1.38·37-s + 1.73·38-s − 1.20·40-s + 1.15·41-s + 0.0351·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.266721883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.266721883\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 5 | \( 1 - 2.49T + 5T^{2} \) |
| 7 | \( 1 - 1.60T + 7T^{2} \) |
| 11 | \( 1 + 0.733T + 11T^{2} \) |
| 13 | \( 1 + 1.89T + 13T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 + 4.63T + 19T^{2} \) |
| 23 | \( 1 - 7.11T + 23T^{2} \) |
| 29 | \( 1 - 0.128T + 29T^{2} \) |
| 37 | \( 1 - 8.42T + 37T^{2} \) |
| 41 | \( 1 - 7.37T + 41T^{2} \) |
| 43 | \( 1 - 0.230T + 43T^{2} \) |
| 47 | \( 1 - 8.03T + 47T^{2} \) |
| 53 | \( 1 + 5.73T + 53T^{2} \) |
| 59 | \( 1 + 9.50T + 59T^{2} \) |
| 61 | \( 1 + 7.84T + 61T^{2} \) |
| 67 | \( 1 - 4.82T + 67T^{2} \) |
| 71 | \( 1 - 3.40T + 71T^{2} \) |
| 73 | \( 1 - 2.69T + 73T^{2} \) |
| 79 | \( 1 - 4.52T + 79T^{2} \) |
| 83 | \( 1 + 2.67T + 83T^{2} \) |
| 89 | \( 1 - 2.20T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−7.79824457654808386182230794114, −7.43647857250154636518580057580, −6.51286716469581367460684846016, −5.97160504500466589321092620629, −5.12374437432092594339364754792, −4.39543075714433784951042627892, −2.97425982728684477423843218336, −2.29240282240830239744965869985, −1.55993113326024540733187035492, −0.73965721044670656144665417298,
0.73965721044670656144665417298, 1.55993113326024540733187035492, 2.29240282240830239744965869985, 2.97425982728684477423843218336, 4.39543075714433784951042627892, 5.12374437432092594339364754792, 5.97160504500466589321092620629, 6.51286716469581367460684846016, 7.43647857250154636518580057580, 7.79824457654808386182230794114
