| L(s) = 1 | − 1.75·2-s − 0.0856·3-s + 1.08·4-s + 0.150·6-s + 1.86·7-s + 1.60·8-s − 2.99·9-s − 1.22·11-s − 0.0929·12-s + 2.40·13-s − 3.26·14-s − 4.99·16-s − 6.05·17-s + 5.25·18-s − 0.159·21-s + 2.15·22-s + 7.20·23-s − 0.137·24-s − 4.22·26-s + 0.513·27-s + 2.02·28-s + 1.28·29-s − 5.87·31-s + 5.55·32-s + 0.104·33-s + 10.6·34-s − 3.24·36-s + ⋯ |
| L(s) = 1 | − 1.24·2-s − 0.0494·3-s + 0.542·4-s + 0.0614·6-s + 0.703·7-s + 0.567·8-s − 0.997·9-s − 0.369·11-s − 0.0268·12-s + 0.667·13-s − 0.873·14-s − 1.24·16-s − 1.46·17-s + 1.23·18-s − 0.0347·21-s + 0.458·22-s + 1.50·23-s − 0.0280·24-s − 0.829·26-s + 0.0987·27-s + 0.381·28-s + 0.238·29-s − 1.05·31-s + 0.982·32-s + 0.0182·33-s + 1.82·34-s − 0.541·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6944774858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6944774858\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.75T + 2T^{2} \) |
| 3 | \( 1 + 0.0856T + 3T^{2} \) |
| 7 | \( 1 - 1.86T + 7T^{2} \) |
| 11 | \( 1 + 1.22T + 11T^{2} \) |
| 13 | \( 1 - 2.40T + 13T^{2} \) |
| 17 | \( 1 + 6.05T + 17T^{2} \) |
| 23 | \( 1 - 7.20T + 23T^{2} \) |
| 29 | \( 1 - 1.28T + 29T^{2} \) |
| 31 | \( 1 + 5.87T + 31T^{2} \) |
| 37 | \( 1 - 1.59T + 37T^{2} \) |
| 41 | \( 1 + 9.72T + 41T^{2} \) |
| 43 | \( 1 - 0.697T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 9.40T + 59T^{2} \) |
| 61 | \( 1 - 3.73T + 61T^{2} \) |
| 67 | \( 1 + 7.41T + 67T^{2} \) |
| 71 | \( 1 + 8.51T + 71T^{2} \) |
| 73 | \( 1 - 6.21T + 73T^{2} \) |
| 79 | \( 1 + 7.39T + 79T^{2} \) |
| 83 | \( 1 + 0.0135T + 83T^{2} \) |
| 89 | \( 1 + 1.25T + 89T^{2} \) |
| 97 | \( 1 + 7.97T + 97T^{2} \) |
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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
−7.981970115359776286210886323582, −7.19087670316789860539766030506, −6.65047851119974744743629144764, −5.70671021475811608179030991347, −4.98187715551303682203450274783, −4.35775732432172182868972609915, −3.26826308874573911756671857068, −2.34108254138263647883533098843, −1.55240966540368217942159873172, −0.50104893645986292055784407962,
0.50104893645986292055784407962, 1.55240966540368217942159873172, 2.34108254138263647883533098843, 3.26826308874573911756671857068, 4.35775732432172182868972609915, 4.98187715551303682203450274783, 5.70671021475811608179030991347, 6.65047851119974744743629144764, 7.19087670316789860539766030506, 7.981970115359776286210886323582
