| L(s) = 1 | + 1.77·2-s − 0.241·3-s + 2.13·4-s + 5-s − 0.426·6-s − 0.709·7-s + 2.01·8-s − 0.941·9-s + 1.77·10-s + 1.49·11-s − 0.514·12-s − 13-s − 1.25·14-s − 0.241·15-s + 1.42·16-s − 1.13·17-s − 1.66·18-s + 2.13·20-s + 0.170·21-s + 2.65·22-s + 1.94·23-s − 0.485·24-s + 25-s − 1.77·26-s + 0.468·27-s − 1.51·28-s − 0.426·30-s + ⋯ |
| L(s) = 1 | + 1.77·2-s − 0.241·3-s + 2.13·4-s + 5-s − 0.426·6-s − 0.709·7-s + 2.01·8-s − 0.941·9-s + 1.77·10-s + 1.49·11-s − 0.514·12-s − 13-s − 1.25·14-s − 0.241·15-s + 1.42·16-s − 1.13·17-s − 1.66·18-s + 2.13·20-s + 0.170·21-s + 2.65·22-s + 1.94·23-s − 0.485·24-s + 25-s − 1.77·26-s + 0.468·27-s − 1.51·28-s − 0.426·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.139761479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.139761479\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 1.77T + T^{2} \) |
| 3 | \( 1 + 0.241T + T^{2} \) |
| 7 | \( 1 + 0.709T + T^{2} \) |
| 11 | \( 1 - 1.49T + T^{2} \) |
| 17 | \( 1 + 1.13T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.94T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.77T + T^{2} \) |
| 47 | \( 1 + 1.94T + T^{2} \) |
| 53 | \( 1 - 1.49T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.49T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 0.709T + T^{2} \) |
| 97 | \( 1 - 0.241T + T^{2} \) |
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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
−9.292014391658777856033638687805, −8.807392771275278954847223213749, −7.09007587341198809330522714171, −6.63488100521241398194754028949, −6.12939132372275766378745388106, −5.19435538235853106664368840107, −4.71595475025130215550375973528, −3.48256531394049166765026733114, −2.83489815514316088580991592171, −1.81549620389973541533802470866,
1.81549620389973541533802470866, 2.83489815514316088580991592171, 3.48256531394049166765026733114, 4.71595475025130215550375973528, 5.19435538235853106664368840107, 6.12939132372275766378745388106, 6.63488100521241398194754028949, 7.09007587341198809330522714171, 8.807392771275278954847223213749, 9.292014391658777856033638687805
