| L(s) = 1 | − 2.54·3-s + 2.36·5-s + 1.43·7-s + 3.45·9-s + 0.0740·11-s − 0.356·13-s − 6.01·15-s − 17-s + 5.87·19-s − 3.63·21-s + 2.37·23-s + 0.609·25-s − 1.14·27-s − 7.93·29-s − 5.12·31-s − 0.188·33-s + 3.38·35-s − 4.00·37-s + 0.904·39-s − 3.80·41-s + 3.55·43-s + 8.17·45-s − 1.51·47-s − 4.95·49-s + 2.54·51-s − 7.98·53-s + 0.175·55-s + ⋯ |
| L(s) = 1 | − 1.46·3-s + 1.05·5-s + 0.540·7-s + 1.15·9-s + 0.0223·11-s − 0.0987·13-s − 1.55·15-s − 0.242·17-s + 1.34·19-s − 0.792·21-s + 0.494·23-s + 0.121·25-s − 0.220·27-s − 1.47·29-s − 0.920·31-s − 0.0327·33-s + 0.572·35-s − 0.658·37-s + 0.144·39-s − 0.594·41-s + 0.542·43-s + 1.21·45-s − 0.220·47-s − 0.707·49-s + 0.355·51-s − 1.09·53-s + 0.0236·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 2.54T + 3T^{2} \) |
| 5 | \( 1 - 2.36T + 5T^{2} \) |
| 7 | \( 1 - 1.43T + 7T^{2} \) |
| 11 | \( 1 - 0.0740T + 11T^{2} \) |
| 13 | \( 1 + 0.356T + 13T^{2} \) |
| 19 | \( 1 - 5.87T + 19T^{2} \) |
| 23 | \( 1 - 2.37T + 23T^{2} \) |
| 29 | \( 1 + 7.93T + 29T^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 + 4.00T + 37T^{2} \) |
| 41 | \( 1 + 3.80T + 41T^{2} \) |
| 43 | \( 1 - 3.55T + 43T^{2} \) |
| 47 | \( 1 + 1.51T + 47T^{2} \) |
| 53 | \( 1 + 7.98T + 53T^{2} \) |
| 61 | \( 1 - 3.91T + 61T^{2} \) |
| 67 | \( 1 + 16.0T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 - 7.16T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 3.07T + 83T^{2} \) |
| 89 | \( 1 - 5.00T + 89T^{2} \) |
| 97 | \( 1 + 1.56T + 97T^{2} \) |
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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
−7.28357755453221094130495962831, −6.66820945273678756161448715361, −5.95047409126635160277261263837, −5.26512603688205901758228798072, −5.19252631982080453068098790496, −4.11556795551754324564603988626, −3.08121833059029673751230165402, −1.89340432717316914129596745369, −1.27862828284806402615648000006, 0,
1.27862828284806402615648000006, 1.89340432717316914129596745369, 3.08121833059029673751230165402, 4.11556795551754324564603988626, 5.19252631982080453068098790496, 5.26512603688205901758228798072, 5.95047409126635160277261263837, 6.66820945273678756161448715361, 7.28357755453221094130495962831
