| L(s) = 1 | − 2.13·2-s − 3.13·3-s + 2.55·4-s + 5-s + 6.69·6-s + 1.41·7-s − 1.18·8-s + 6.84·9-s − 2.13·10-s − 1.68·11-s − 8.01·12-s − 4.54·13-s − 3.02·14-s − 3.13·15-s − 2.58·16-s − 3.25·17-s − 14.6·18-s − 0.0290·19-s + 2.55·20-s − 4.44·21-s + 3.60·22-s + 2.28·23-s + 3.71·24-s + 25-s + 9.69·26-s − 12.0·27-s + 3.61·28-s + ⋯ |
| L(s) = 1 | − 1.50·2-s − 1.81·3-s + 1.27·4-s + 0.447·5-s + 2.73·6-s + 0.535·7-s − 0.418·8-s + 2.28·9-s − 0.674·10-s − 0.509·11-s − 2.31·12-s − 1.25·13-s − 0.808·14-s − 0.810·15-s − 0.645·16-s − 0.790·17-s − 3.44·18-s − 0.00665·19-s + 0.571·20-s − 0.970·21-s + 0.768·22-s + 0.476·23-s + 0.758·24-s + 0.200·25-s + 1.90·26-s − 2.31·27-s + 0.684·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 2.13T + 2T^{2} \) |
| 3 | \( 1 + 3.13T + 3T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 + 1.68T + 11T^{2} \) |
| 13 | \( 1 + 4.54T + 13T^{2} \) |
| 17 | \( 1 + 3.25T + 17T^{2} \) |
| 19 | \( 1 + 0.0290T + 19T^{2} \) |
| 23 | \( 1 - 2.28T + 23T^{2} \) |
| 29 | \( 1 - 8.95T + 29T^{2} \) |
| 31 | \( 1 - 4.17T + 31T^{2} \) |
| 37 | \( 1 - 0.842T + 37T^{2} \) |
| 41 | \( 1 - 1.39T + 41T^{2} \) |
| 43 | \( 1 + 9.59T + 43T^{2} \) |
| 47 | \( 1 + 4.29T + 47T^{2} \) |
| 53 | \( 1 - 7.25T + 53T^{2} \) |
| 59 | \( 1 + 1.64T + 59T^{2} \) |
| 61 | \( 1 + 2.40T + 61T^{2} \) |
| 67 | \( 1 - 1.37T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 1.36T + 79T^{2} \) |
| 83 | \( 1 - 1.41T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−9.640403417030194673129516639309, −8.570072655819168419561951542619, −7.66632037690591895493470182050, −6.85546431084953903009977219493, −6.29091079946606590390969027126, −4.97526153808225759590783620167, −4.73146117854039170895714423881, −2.37804956399506050881238410061, −1.18934553003241026309313664719, 0,
1.18934553003241026309313664719, 2.37804956399506050881238410061, 4.73146117854039170895714423881, 4.97526153808225759590783620167, 6.29091079946606590390969027126, 6.85546431084953903009977219493, 7.66632037690591895493470182050, 8.570072655819168419561951542619, 9.640403417030194673129516639309
