| L(s) = 1 | + 2-s + 2.78·3-s + 4-s + 5-s + 2.78·6-s − 0.104·7-s + 8-s + 4.77·9-s + 10-s + 1.61·11-s + 2.78·12-s − 0.820·13-s − 0.104·14-s + 2.78·15-s + 16-s + 2.39·17-s + 4.77·18-s − 5.04·19-s + 20-s − 0.292·21-s + 1.61·22-s + 2.78·24-s + 25-s − 0.820·26-s + 4.94·27-s − 0.104·28-s + 9.29·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.60·3-s + 0.5·4-s + 0.447·5-s + 1.13·6-s − 0.0396·7-s + 0.353·8-s + 1.59·9-s + 0.316·10-s + 0.486·11-s + 0.804·12-s − 0.227·13-s − 0.0280·14-s + 0.719·15-s + 0.250·16-s + 0.580·17-s + 1.12·18-s − 1.15·19-s + 0.223·20-s − 0.0637·21-s + 0.343·22-s + 0.569·24-s + 0.200·25-s − 0.160·26-s + 0.952·27-s − 0.0198·28-s + 1.72·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.628156851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.628156851\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2.78T + 3T^{2} \) |
| 7 | \( 1 + 0.104T + 7T^{2} \) |
| 11 | \( 1 - 1.61T + 11T^{2} \) |
| 13 | \( 1 + 0.820T + 13T^{2} \) |
| 17 | \( 1 - 2.39T + 17T^{2} \) |
| 19 | \( 1 + 5.04T + 19T^{2} \) |
| 29 | \( 1 - 9.29T + 29T^{2} \) |
| 31 | \( 1 - 6.93T + 31T^{2} \) |
| 37 | \( 1 + 3.78T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 1.06T + 43T^{2} \) |
| 47 | \( 1 + 8.70T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 1.88T + 59T^{2} \) |
| 61 | \( 1 - 0.718T + 61T^{2} \) |
| 67 | \( 1 + 3.50T + 67T^{2} \) |
| 71 | \( 1 + 1.31T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−8.130874679164845239820769764472, −7.67082492614831538574842424724, −6.51125132900174147736064964838, −6.36324146236970734881721334661, −5.00606886510720568741342361148, −4.41699912930090245932956465979, −3.56247648996091124668270086029, −2.87861418965640232225666755119, −2.22693072839910045397360817205, −1.29898332226448899185066023879,
1.29898332226448899185066023879, 2.22693072839910045397360817205, 2.87861418965640232225666755119, 3.56247648996091124668270086029, 4.41699912930090245932956465979, 5.00606886510720568741342361148, 6.36324146236970734881721334661, 6.51125132900174147736064964838, 7.67082492614831538574842424724, 8.130874679164845239820769764472
