| L(s) = 1 | − 2-s − 2.64·3-s + 4-s + 3.64·5-s + 2.64·6-s − 8-s + 4.00·9-s − 3.64·10-s + 11-s − 2.64·12-s + 5·13-s − 9.64·15-s + 16-s + 6·17-s − 4.00·18-s − 0.354·19-s + 3.64·20-s − 22-s − 3.64·23-s + 2.64·24-s + 8.29·25-s − 5·26-s − 2.64·27-s − 4.29·29-s + 9.64·30-s − 4·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.52·3-s + 0.5·4-s + 1.63·5-s + 1.08·6-s − 0.353·8-s + 1.33·9-s − 1.15·10-s + 0.301·11-s − 0.763·12-s + 1.38·13-s − 2.49·15-s + 0.250·16-s + 1.45·17-s − 0.942·18-s − 0.0812·19-s + 0.815·20-s − 0.213·22-s − 0.760·23-s + 0.540·24-s + 1.65·25-s − 0.980·26-s − 0.509·27-s − 0.796·29-s + 1.76·30-s − 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.038319442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.038319442\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2.64T + 3T^{2} \) |
| 5 | \( 1 - 3.64T + 5T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 0.354T + 19T^{2} \) |
| 23 | \( 1 + 3.64T + 23T^{2} \) |
| 29 | \( 1 + 4.29T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 1.64T + 37T^{2} \) |
| 41 | \( 1 + 4.93T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 + 3.64T + 53T^{2} \) |
| 59 | \( 1 + 0.645T + 59T^{2} \) |
| 61 | \( 1 - 3.70T + 61T^{2} \) |
| 67 | \( 1 - 3.93T + 67T^{2} \) |
| 71 | \( 1 - 9.64T + 71T^{2} \) |
| 73 | \( 1 - 5.64T + 73T^{2} \) |
| 79 | \( 1 - 2.64T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 5.70T + 97T^{2} \) |
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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
−9.985039884127982784318311915840, −9.339689780225081188860287075261, −8.362901087720273457371685153232, −7.13670645228546176268909152628, −6.25446640921290216886966184881, −5.82144941913944127481334930828, −5.21888988757620254684977205127, −3.61912807627275469830733243634, −1.91107383944739421659823034891, −1.00798817726345975032766867908,
1.00798817726345975032766867908, 1.91107383944739421659823034891, 3.61912807627275469830733243634, 5.21888988757620254684977205127, 5.82144941913944127481334930828, 6.25446640921290216886966184881, 7.13670645228546176268909152628, 8.362901087720273457371685153232, 9.339689780225081188860287075261, 9.985039884127982784318311915840
