| L(s) = 1 | − 2.33·3-s + 8.20·5-s − 7·7-s − 21.5·9-s − 60.9·11-s − 75.6·13-s − 19.1·15-s + 8.09·17-s − 116.·19-s + 16.3·21-s + 16.0·23-s − 57.6·25-s + 113.·27-s + 155.·29-s + 242.·31-s + 142.·33-s − 57.4·35-s − 94.5·37-s + 176.·39-s + 41·41-s + 252.·43-s − 176.·45-s + 519.·47-s + 49·49-s − 18.9·51-s − 19.9·53-s − 500.·55-s + ⋯ |
| L(s) = 1 | − 0.449·3-s + 0.734·5-s − 0.377·7-s − 0.797·9-s − 1.67·11-s − 1.61·13-s − 0.330·15-s + 0.115·17-s − 1.40·19-s + 0.170·21-s + 0.145·23-s − 0.461·25-s + 0.808·27-s + 0.994·29-s + 1.40·31-s + 0.751·33-s − 0.277·35-s − 0.420·37-s + 0.725·39-s + 0.156·41-s + 0.894·43-s − 0.585·45-s + 1.61·47-s + 0.142·49-s − 0.0519·51-s − 0.0517·53-s − 1.22·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8125095256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8125095256\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 41 | \( 1 - 41T \) |
| good | 3 | \( 1 + 2.33T + 27T^{2} \) |
| 5 | \( 1 - 8.20T + 125T^{2} \) |
| 11 | \( 1 + 60.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 75.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 8.09T + 4.91e3T^{2} \) |
| 19 | \( 1 + 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 16.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 155.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 242.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 94.5T + 5.06e4T^{2} \) |
| 43 | \( 1 - 252.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 519.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 19.9T + 1.48e5T^{2} \) |
| 59 | \( 1 - 269.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 58.3T + 2.26e5T^{2} \) |
| 67 | \( 1 - 371.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 558.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 446.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 260.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 887.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 360.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.62e3T + 9.12e5T^{2} \) |
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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
−9.599181833848940461762762294882, −8.580408239087378991521395768645, −7.81026385295334647631130485878, −6.81283441074672124449308691306, −5.92295811845469523602638045351, −5.28978665599140375184213005921, −4.46294751627369857548867709579, −2.72093533586734739864643689922, −2.38751026241261958607458242993, −0.44144433384927007377797886024,
0.44144433384927007377797886024, 2.38751026241261958607458242993, 2.72093533586734739864643689922, 4.46294751627369857548867709579, 5.28978665599140375184213005921, 5.92295811845469523602638045351, 6.81283441074672124449308691306, 7.81026385295334647631130485878, 8.580408239087378991521395768645, 9.599181833848940461762762294882
