| L(s) = 1 | + 2-s − 2.23·3-s + 4-s + 0.812·5-s − 2.23·6-s + 7-s + 8-s + 1.99·9-s + 0.812·10-s + 3.26·11-s − 2.23·12-s + 2.96·13-s + 14-s − 1.81·15-s + 16-s − 2.78·17-s + 1.99·18-s − 4.32·19-s + 0.812·20-s − 2.23·21-s + 3.26·22-s − 8.63·23-s − 2.23·24-s − 4.33·25-s + 2.96·26-s + 2.24·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.29·3-s + 0.5·4-s + 0.363·5-s − 0.912·6-s + 0.377·7-s + 0.353·8-s + 0.664·9-s + 0.256·10-s + 0.984·11-s − 0.645·12-s + 0.821·13-s + 0.267·14-s − 0.468·15-s + 0.250·16-s − 0.675·17-s + 0.470·18-s − 0.991·19-s + 0.181·20-s − 0.487·21-s + 0.696·22-s − 1.80·23-s − 0.456·24-s − 0.867·25-s + 0.581·26-s + 0.432·27-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 | \( 1 + 2.23T + 3T^{2} \) |
| 5 | \( 1 - 0.812T + 5T^{2} \) |
| 11 | \( 1 - 3.26T + 11T^{2} \) |
| 13 | \( 1 - 2.96T + 13T^{2} \) |
| 17 | \( 1 + 2.78T + 17T^{2} \) |
| 19 | \( 1 + 4.32T + 19T^{2} \) |
| 23 | \( 1 + 8.63T + 23T^{2} \) |
| 29 | \( 1 + 9.37T + 29T^{2} \) |
| 31 | \( 1 - 5.70T + 31T^{2} \) |
| 37 | \( 1 - 7.34T + 37T^{2} \) |
| 41 | \( 1 - 1.76T + 41T^{2} \) |
| 43 | \( 1 + 1.70T + 43T^{2} \) |
| 47 | \( 1 + 9.53T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 + 8.71T + 59T^{2} \) |
| 61 | \( 1 + 4.63T + 61T^{2} \) |
| 67 | \( 1 - 9.29T + 67T^{2} \) |
| 71 | \( 1 + 16.1T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 9.75T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 6.52T + 89T^{2} \) |
| 97 | \( 1 + 1.25T + 97T^{2} \) |
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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
−7.59731302139646937881359120829, −6.45690141341754267760184910913, −6.18116884944776035660510850315, −5.85147937903368415659536413560, −4.75249342261489453020208645708, −4.29538540091802099562148406136, −3.51973542680168501040302667385, −2.13246259951548674004611149585, −1.45358169551198105087466964820, 0,
1.45358169551198105087466964820, 2.13246259951548674004611149585, 3.51973542680168501040302667385, 4.29538540091802099562148406136, 4.75249342261489453020208645708, 5.85147937903368415659536413560, 6.18116884944776035660510850315, 6.45690141341754267760184910913, 7.59731302139646937881359120829
