L(s) = 1 | + 2-s + 2.93·3-s + 4-s − 5-s + 2.93·6-s − 1.31·7-s + 8-s + 5.63·9-s − 10-s − 0.258·11-s + 2.93·12-s − 5.87·13-s − 1.31·14-s − 2.93·15-s + 16-s + 4.25·17-s + 5.63·18-s + 2.93·19-s − 20-s − 3.87·21-s − 0.258·22-s − 8.51·23-s + 2.93·24-s + 25-s − 5.87·26-s + 7.75·27-s − 1.31·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.69·3-s + 0.5·4-s − 0.447·5-s + 1.19·6-s − 0.498·7-s + 0.353·8-s + 1.87·9-s − 0.316·10-s − 0.0780·11-s + 0.848·12-s − 1.63·13-s − 0.352·14-s − 0.758·15-s + 0.250·16-s + 1.03·17-s + 1.32·18-s + 0.674·19-s − 0.223·20-s − 0.846·21-s − 0.0551·22-s − 1.77·23-s + 0.599·24-s + 0.200·25-s − 1.15·26-s + 1.49·27-s − 0.249·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.927119715\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.927119715\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 - 2.93T + 3T^{2} \) |
| 7 | \( 1 + 1.31T + 7T^{2} \) |
| 11 | \( 1 + 0.258T + 11T^{2} \) |
| 13 | \( 1 + 5.87T + 13T^{2} \) |
| 17 | \( 1 - 4.25T + 17T^{2} \) |
| 19 | \( 1 - 2.93T + 19T^{2} \) |
| 23 | \( 1 + 8.51T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 - 3.95T + 31T^{2} \) |
| 41 | \( 1 + 8.89T + 41T^{2} \) |
| 43 | \( 1 + 5.61T + 43T^{2} \) |
| 47 | \( 1 - 5.57T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 - 9.57T + 59T^{2} \) |
| 61 | \( 1 - 0.380T + 61T^{2} \) |
| 67 | \( 1 - 5.57T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 2.51T + 73T^{2} \) |
| 79 | \( 1 + 7.69T + 79T^{2} \) |
| 83 | \( 1 + 6.17T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−11.82636435869868264500840212163, −9.996826795763793106699149854426, −9.815362902437729114919347854157, −8.404430912473195544754393272519, −7.67374705854895532338206160241, −6.93467493168068283699568623376, −5.32531411687525139171609671547, −4.02831704671153362193075685414, −3.20876494903941789673814290933, −2.19114773505198629580624255052,
2.19114773505198629580624255052, 3.20876494903941789673814290933, 4.02831704671153362193075685414, 5.32531411687525139171609671547, 6.93467493168068283699568623376, 7.67374705854895532338206160241, 8.404430912473195544754393272519, 9.815362902437729114919347854157, 9.996826795763793106699149854426, 11.82636435869868264500840212163