| L(s) = 1 | − 0.518·2-s − 1.73·4-s + 4.37·5-s − 5.00·7-s + 1.93·8-s − 2.26·10-s + 1.38·11-s + 3.17·13-s + 2.59·14-s + 2.45·16-s − 4.31·17-s − 1.85·19-s − 7.57·20-s − 0.719·22-s + 5.62·23-s + 14.1·25-s − 1.64·26-s + 8.67·28-s + 5.54·29-s − 6.48·31-s − 5.14·32-s + 2.23·34-s − 21.9·35-s + 7.44·37-s + 0.960·38-s + 8.46·40-s + 3.29·41-s + ⋯ |
| L(s) = 1 | − 0.366·2-s − 0.865·4-s + 1.95·5-s − 1.89·7-s + 0.684·8-s − 0.717·10-s + 0.418·11-s + 0.881·13-s + 0.694·14-s + 0.614·16-s − 1.04·17-s − 0.424·19-s − 1.69·20-s − 0.153·22-s + 1.17·23-s + 2.82·25-s − 0.323·26-s + 1.63·28-s + 1.02·29-s − 1.16·31-s − 0.909·32-s + 0.383·34-s − 3.70·35-s + 1.22·37-s + 0.155·38-s + 1.33·40-s + 0.514·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.323163007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.323163007\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 0.518T + 2T^{2} \) |
| 5 | \( 1 - 4.37T + 5T^{2} \) |
| 7 | \( 1 + 5.00T + 7T^{2} \) |
| 11 | \( 1 - 1.38T + 11T^{2} \) |
| 13 | \( 1 - 3.17T + 13T^{2} \) |
| 17 | \( 1 + 4.31T + 17T^{2} \) |
| 19 | \( 1 + 1.85T + 19T^{2} \) |
| 23 | \( 1 - 5.62T + 23T^{2} \) |
| 29 | \( 1 - 5.54T + 29T^{2} \) |
| 31 | \( 1 + 6.48T + 31T^{2} \) |
| 37 | \( 1 - 7.44T + 37T^{2} \) |
| 41 | \( 1 - 3.29T + 41T^{2} \) |
| 43 | \( 1 + 1.77T + 43T^{2} \) |
| 47 | \( 1 - 9.45T + 47T^{2} \) |
| 53 | \( 1 + 1.98T + 53T^{2} \) |
| 59 | \( 1 + 3.89T + 59T^{2} \) |
| 61 | \( 1 - 3.80T + 61T^{2} \) |
| 67 | \( 1 + 6.46T + 67T^{2} \) |
| 71 | \( 1 - 4.99T + 71T^{2} \) |
| 73 | \( 1 + 0.636T + 73T^{2} \) |
| 79 | \( 1 - 6.86T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 + 17.7T + 89T^{2} \) |
| 97 | \( 1 - 8.19T + 97T^{2} \) |
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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
−9.622431353099411385232307504474, −9.068799112832725595996200097765, −8.796685984465292228070993378505, −7.07428506991894251541955741159, −6.29708680304006058458766743756, −5.86188726954771064649986266205, −4.69633203081484792975321484476, −3.49513967504006243161602689588, −2.40012420658748867140992550316, −0.953821261243499088687019134979,
0.953821261243499088687019134979, 2.40012420658748867140992550316, 3.49513967504006243161602689588, 4.69633203081484792975321484476, 5.86188726954771064649986266205, 6.29708680304006058458766743756, 7.07428506991894251541955741159, 8.796685984465292228070993378505, 9.068799112832725595996200097765, 9.622431353099411385232307504474
