| L(s) = 1 | + 0.931·2-s − 1.13·4-s + 5-s − 1.36·7-s − 2.91·8-s + 0.931·10-s − 4.39·11-s + 13-s − 1.27·14-s − 0.450·16-s + 5.08·17-s + 1.31·19-s − 1.13·20-s − 4.09·22-s + 5.51·23-s + 25-s + 0.931·26-s + 1.54·28-s + 5.36·29-s + 8.54·31-s + 5.41·32-s + 4.73·34-s − 1.36·35-s − 2.75·37-s + 1.22·38-s − 2.91·40-s + 11.0·41-s + ⋯ |
| L(s) = 1 | + 0.658·2-s − 0.566·4-s + 0.447·5-s − 0.516·7-s − 1.03·8-s + 0.294·10-s − 1.32·11-s + 0.277·13-s − 0.340·14-s − 0.112·16-s + 1.23·17-s + 0.302·19-s − 0.253·20-s − 0.873·22-s + 1.15·23-s + 0.200·25-s + 0.182·26-s + 0.292·28-s + 0.997·29-s + 1.53·31-s + 0.957·32-s + 0.811·34-s − 0.231·35-s − 0.453·37-s + 0.198·38-s − 0.461·40-s + 1.72·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.885948001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.885948001\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 0.931T + 2T^{2} \) |
| 7 | \( 1 + 1.36T + 7T^{2} \) |
| 11 | \( 1 + 4.39T + 11T^{2} \) |
| 17 | \( 1 - 5.08T + 17T^{2} \) |
| 19 | \( 1 - 1.31T + 19T^{2} \) |
| 23 | \( 1 - 5.51T + 23T^{2} \) |
| 29 | \( 1 - 5.36T + 29T^{2} \) |
| 31 | \( 1 - 8.54T + 31T^{2} \) |
| 37 | \( 1 + 2.75T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 4.32T + 43T^{2} \) |
| 47 | \( 1 + 7.23T + 47T^{2} \) |
| 53 | \( 1 + 3.52T + 53T^{2} \) |
| 59 | \( 1 + 3.42T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 - 0.887T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 - 5.15T + 73T^{2} \) |
| 79 | \( 1 - 7.01T + 79T^{2} \) |
| 83 | \( 1 + 4.94T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−9.457067559464364225599096023655, −8.477265618002185381543425886138, −7.82759073732434831622086254748, −6.69997936425466844789717020399, −5.85071947819275140561690063796, −5.22603334547831554577154089525, −4.48630136320708765664168511864, −3.23615311009934225157001882099, −2.74177013489687379517076294885, −0.866439235850902683195044839963,
0.866439235850902683195044839963, 2.74177013489687379517076294885, 3.23615311009934225157001882099, 4.48630136320708765664168511864, 5.22603334547831554577154089525, 5.85071947819275140561690063796, 6.69997936425466844789717020399, 7.82759073732434831622086254748, 8.477265618002185381543425886138, 9.457067559464364225599096023655
