| L(s) = 1 | − 2-s + 0.814·3-s + 4-s + 0.275·5-s − 0.814·6-s + 7-s − 8-s − 2.33·9-s − 0.275·10-s − 0.495·11-s + 0.814·12-s + 0.918·13-s − 14-s + 0.224·15-s + 16-s − 3.76·17-s + 2.33·18-s + 4.78·19-s + 0.275·20-s + 0.814·21-s + 0.495·22-s − 2.19·23-s − 0.814·24-s − 4.92·25-s − 0.918·26-s − 4.34·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.470·3-s + 0.5·4-s + 0.123·5-s − 0.332·6-s + 0.377·7-s − 0.353·8-s − 0.778·9-s − 0.0870·10-s − 0.149·11-s + 0.235·12-s + 0.254·13-s − 0.267·14-s + 0.0578·15-s + 0.250·16-s − 0.913·17-s + 0.550·18-s + 1.09·19-s + 0.0615·20-s + 0.177·21-s + 0.105·22-s − 0.457·23-s − 0.166·24-s − 0.984·25-s − 0.180·26-s − 0.836·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 - 0.814T + 3T^{2} \) |
| 5 | \( 1 - 0.275T + 5T^{2} \) |
| 11 | \( 1 + 0.495T + 11T^{2} \) |
| 13 | \( 1 - 0.918T + 13T^{2} \) |
| 17 | \( 1 + 3.76T + 17T^{2} \) |
| 19 | \( 1 - 4.78T + 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 + 0.158T + 29T^{2} \) |
| 31 | \( 1 - 9.95T + 31T^{2} \) |
| 37 | \( 1 - 1.30T + 37T^{2} \) |
| 41 | \( 1 + 9.73T + 41T^{2} \) |
| 43 | \( 1 - 1.46T + 43T^{2} \) |
| 47 | \( 1 + 5.94T + 47T^{2} \) |
| 53 | \( 1 - 1.04T + 53T^{2} \) |
| 59 | \( 1 - 4.67T + 59T^{2} \) |
| 61 | \( 1 - 3.90T + 61T^{2} \) |
| 67 | \( 1 + 1.29T + 67T^{2} \) |
| 71 | \( 1 + 6.58T + 71T^{2} \) |
| 73 | \( 1 + 4.35T + 73T^{2} \) |
| 79 | \( 1 + 5.47T + 79T^{2} \) |
| 83 | \( 1 + 2.20T + 83T^{2} \) |
| 89 | \( 1 + 7.84T + 89T^{2} \) |
| 97 | \( 1 + 5.03T + 97T^{2} \) |
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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
−7.977614942864286837022483185072, −7.18890996819841570490635444335, −6.37748654519226248302832379609, −5.71233742784720513811481090100, −4.89296501785051340272230553093, −3.89378059891207534286429517402, −2.98785897264455743938461512276, −2.29896096057660297604919743017, −1.34562696471383248472831835965, 0,
1.34562696471383248472831835965, 2.29896096057660297604919743017, 2.98785897264455743938461512276, 3.89378059891207534286429517402, 4.89296501785051340272230553093, 5.71233742784720513811481090100, 6.37748654519226248302832379609, 7.18890996819841570490635444335, 7.977614942864286837022483185072
