| L(s) = 1 | − 2-s + 4-s − 5-s + 1.61·7-s − 8-s + 10-s − 3.85·11-s + 4.09·13-s − 1.61·14-s + 16-s + 5.09·17-s − 4.85·19-s − 20-s + 3.85·22-s − 23-s + 25-s − 4.09·26-s + 1.61·28-s + 4.76·29-s − 2.09·31-s − 32-s − 5.09·34-s − 1.61·35-s − 2.47·37-s + 4.85·38-s + 40-s + 12.3·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.611·7-s − 0.353·8-s + 0.316·10-s − 1.16·11-s + 1.13·13-s − 0.432·14-s + 0.250·16-s + 1.23·17-s − 1.11·19-s − 0.223·20-s + 0.821·22-s − 0.208·23-s + 0.200·25-s − 0.802·26-s + 0.305·28-s + 0.884·29-s − 0.375·31-s − 0.176·32-s − 0.872·34-s − 0.273·35-s − 0.406·37-s + 0.787·38-s + 0.158·40-s + 1.92·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.170604794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.170604794\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 1.61T + 7T^{2} \) |
| 11 | \( 1 + 3.85T + 11T^{2} \) |
| 13 | \( 1 - 4.09T + 13T^{2} \) |
| 17 | \( 1 - 5.09T + 17T^{2} \) |
| 19 | \( 1 + 4.85T + 19T^{2} \) |
| 29 | \( 1 - 4.76T + 29T^{2} \) |
| 31 | \( 1 + 2.09T + 31T^{2} \) |
| 37 | \( 1 + 2.47T + 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 9.70T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 6.32T + 61T^{2} \) |
| 67 | \( 1 - 5.52T + 67T^{2} \) |
| 71 | \( 1 + 7.09T + 71T^{2} \) |
| 73 | \( 1 + 1.23T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 1.52T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−8.915919089267394368139024097000, −8.193832656684335668977110465955, −7.919274553111697184622416661461, −6.95851054108973961374312161214, −6.00760005514998738072551903495, −5.22464210615257006858018019389, −4.15340517730267265336944226327, −3.15578636193988013655177999170, −2.05326656139358273818644960548, −0.803817388064460988594638353418,
0.803817388064460988594638353418, 2.05326656139358273818644960548, 3.15578636193988013655177999170, 4.15340517730267265336944226327, 5.22464210615257006858018019389, 6.00760005514998738072551903495, 6.95851054108973961374312161214, 7.919274553111697184622416661461, 8.193832656684335668977110465955, 8.915919089267394368139024097000
