| L(s) = 1 | − 2.22·2-s + 3-s + 2.97·4-s + 0.537·5-s − 2.22·6-s − 1.88·7-s − 2.16·8-s + 9-s − 1.19·10-s + 1.11·11-s + 2.97·12-s + 1.56·13-s + 4.19·14-s + 0.537·15-s − 1.11·16-s − 3.41·17-s − 2.22·18-s − 2.62·19-s + 1.59·20-s − 1.88·21-s − 2.49·22-s + 23-s − 2.16·24-s − 4.71·25-s − 3.49·26-s + 27-s − 5.59·28-s + ⋯ |
| L(s) = 1 | − 1.57·2-s + 0.577·3-s + 1.48·4-s + 0.240·5-s − 0.910·6-s − 0.711·7-s − 0.765·8-s + 0.333·9-s − 0.378·10-s + 0.336·11-s + 0.857·12-s + 0.434·13-s + 1.12·14-s + 0.138·15-s − 0.278·16-s − 0.828·17-s − 0.525·18-s − 0.602·19-s + 0.356·20-s − 0.410·21-s − 0.531·22-s + 0.208·23-s − 0.442·24-s − 0.942·25-s − 0.685·26-s + 0.192·27-s − 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.22T + 2T^{2} \) |
| 5 | \( 1 - 0.537T + 5T^{2} \) |
| 7 | \( 1 + 1.88T + 7T^{2} \) |
| 11 | \( 1 - 1.11T + 11T^{2} \) |
| 13 | \( 1 - 1.56T + 13T^{2} \) |
| 17 | \( 1 + 3.41T + 17T^{2} \) |
| 19 | \( 1 + 2.62T + 19T^{2} \) |
| 31 | \( 1 + 2.26T + 31T^{2} \) |
| 37 | \( 1 - 1.64T + 37T^{2} \) |
| 41 | \( 1 - 5.06T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + 6.06T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 8.80T + 59T^{2} \) |
| 61 | \( 1 + 3.73T + 61T^{2} \) |
| 67 | \( 1 + 7.67T + 67T^{2} \) |
| 71 | \( 1 + 2.43T + 71T^{2} \) |
| 73 | \( 1 - 1.02T + 73T^{2} \) |
| 79 | \( 1 - 1.24T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + 7.41T + 97T^{2} \) |
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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
−8.833852153293619342815041258429, −8.257885135266819077463183479716, −7.41327097175532181819858586103, −6.66791525650015669515014085182, −6.03333691972207041370487311060, −4.55044108040538804185647944625, −3.50301651376845247222701889284, −2.38272842662696696594179343753, −1.50388471664017327210206339184, 0,
1.50388471664017327210206339184, 2.38272842662696696594179343753, 3.50301651376845247222701889284, 4.55044108040538804185647944625, 6.03333691972207041370487311060, 6.66791525650015669515014085182, 7.41327097175532181819858586103, 8.257885135266819077463183479716, 8.833852153293619342815041258429
