| L(s) = 1 | + 9.38·3-s + 14.2·5-s − 7·7-s + 61.1·9-s + 4.55·11-s + 1.77·13-s + 134.·15-s + 35.9·17-s + 17.5·19-s − 65.7·21-s − 62.5·23-s + 79.0·25-s + 320.·27-s + 263.·29-s + 74.3·31-s + 42.7·33-s − 99.9·35-s − 352.·37-s + 16.6·39-s + 41·41-s − 67.3·43-s + 873.·45-s + 145.·47-s + 49·49-s + 337.·51-s + 179.·53-s + 65.0·55-s + ⋯ |
| L(s) = 1 | + 1.80·3-s + 1.27·5-s − 0.377·7-s + 2.26·9-s + 0.124·11-s + 0.0378·13-s + 2.30·15-s + 0.513·17-s + 0.211·19-s − 0.682·21-s − 0.567·23-s + 0.632·25-s + 2.28·27-s + 1.68·29-s + 0.430·31-s + 0.225·33-s − 0.482·35-s − 1.56·37-s + 0.0684·39-s + 0.156·41-s − 0.238·43-s + 2.89·45-s + 0.450·47-s + 0.142·49-s + 0.927·51-s + 0.464·53-s + 0.159·55-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
| \(L(2)\) |
\(\approx\) |
\(5.676637100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.676637100\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 9.38T + 27T^{2} \) |
| 5 | \( 1 - 14.2T + 125T^{2} \) |
| 11 | \( 1 - 4.55T + 1.33e3T^{2} \) |
| 13 | \( 1 - 1.77T + 2.19e3T^{2} \) |
| 17 | \( 1 - 35.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 17.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 62.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 263.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 74.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 352.T + 5.06e4T^{2} \) |
| 43 | \( 1 + 67.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 145.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 179.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 313.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 198.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 436.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 648.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 408.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 351.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 773.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 177.T + 9.12e5T^{2} \) |
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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
−9.407588643157412358266005230933, −8.688997963358145131073483692400, −8.008437536222838011720251953610, −7.01442784791158609672624493431, −6.22758507716722453801069413599, −5.07295805550120634797346091935, −3.88608069926835842974580840955, −2.97805195885550519876687776655, −2.22261213235441130699811470627, −1.27251770940532284907386113469,
1.27251770940532284907386113469, 2.22261213235441130699811470627, 2.97805195885550519876687776655, 3.88608069926835842974580840955, 5.07295805550120634797346091935, 6.22758507716722453801069413599, 7.01442784791158609672624493431, 8.008437536222838011720251953610, 8.688997963358145131073483692400, 9.407588643157412358266005230933
