| L(s) = 1 | − 0.526·2-s − 2.87·3-s − 1.72·4-s + 5-s + 1.51·6-s − 4.16·7-s + 1.95·8-s + 5.28·9-s − 0.526·10-s + 0.133·11-s + 4.95·12-s − 0.0230·13-s + 2.19·14-s − 2.87·15-s + 2.41·16-s − 0.577·17-s − 2.77·18-s + 4.83·19-s − 1.72·20-s + 11.9·21-s − 0.0700·22-s + 2.38·23-s − 5.63·24-s + 25-s + 0.0121·26-s − 6.56·27-s + 7.17·28-s + ⋯ |
| L(s) = 1 | − 0.371·2-s − 1.66·3-s − 0.861·4-s + 0.447·5-s + 0.618·6-s − 1.57·7-s + 0.692·8-s + 1.76·9-s − 0.166·10-s + 0.0401·11-s + 1.43·12-s − 0.00638·13-s + 0.585·14-s − 0.743·15-s + 0.603·16-s − 0.140·17-s − 0.654·18-s + 1.10·19-s − 0.385·20-s + 2.61·21-s − 0.0149·22-s + 0.497·23-s − 1.15·24-s + 0.200·25-s + 0.00237·26-s − 1.26·27-s + 1.35·28-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 | \( 1 + 0.526T + 2T^{2} \) |
| 3 | \( 1 + 2.87T + 3T^{2} \) |
| 7 | \( 1 + 4.16T + 7T^{2} \) |
| 11 | \( 1 - 0.133T + 11T^{2} \) |
| 13 | \( 1 + 0.0230T + 13T^{2} \) |
| 17 | \( 1 + 0.577T + 17T^{2} \) |
| 19 | \( 1 - 4.83T + 19T^{2} \) |
| 23 | \( 1 - 2.38T + 23T^{2} \) |
| 29 | \( 1 - 5.43T + 29T^{2} \) |
| 31 | \( 1 + 6.98T + 31T^{2} \) |
| 37 | \( 1 - 6.76T + 37T^{2} \) |
| 41 | \( 1 - 0.889T + 41T^{2} \) |
| 43 | \( 1 - 1.16T + 43T^{2} \) |
| 47 | \( 1 + 7.26T + 47T^{2} \) |
| 53 | \( 1 + 8.72T + 53T^{2} \) |
| 59 | \( 1 - 4.97T + 59T^{2} \) |
| 61 | \( 1 - 6.87T + 61T^{2} \) |
| 67 | \( 1 + 8.47T + 67T^{2} \) |
| 71 | \( 1 + 6.49T + 71T^{2} \) |
| 73 | \( 1 + 3.09T + 73T^{2} \) |
| 79 | \( 1 + 2.01T + 79T^{2} \) |
| 83 | \( 1 + 3.73T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 + 7.50T + 97T^{2} \) |
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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
−9.715414590777815077677656032599, −8.808593098594758950910147286415, −7.46073647321231973008675680230, −6.67815868798649902234728724230, −5.91918153210982325879639889404, −5.25912409915089176044351510864, −4.34161608314483451982189363721, −3.15796462785691752403581448182, −1.12657699824633115040194860108, 0,
1.12657699824633115040194860108, 3.15796462785691752403581448182, 4.34161608314483451982189363721, 5.25912409915089176044351510864, 5.91918153210982325879639889404, 6.67815868798649902234728724230, 7.46073647321231973008675680230, 8.808593098594758950910147286415, 9.715414590777815077677656032599
