| L(s) = 1 | + 3-s + 3.29·5-s + 2·7-s + 9-s − 0.786·11-s − 1.29·13-s + 3.29·15-s − 4.08·17-s + 6.87·19-s + 2·21-s + 8.16·23-s + 5.87·25-s + 27-s − 5.80·29-s − 10.2·31-s − 0.786·33-s + 6.59·35-s − 6.95·37-s − 1.29·39-s − 41-s − 3.38·43-s + 3.29·45-s + 1.21·47-s − 3·49-s − 4.08·51-s − 5.02·53-s − 2.59·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.47·5-s + 0.755·7-s + 0.333·9-s − 0.237·11-s − 0.359·13-s + 0.851·15-s − 0.990·17-s + 1.57·19-s + 0.436·21-s + 1.70·23-s + 1.17·25-s + 0.192·27-s − 1.07·29-s − 1.84·31-s − 0.136·33-s + 1.11·35-s − 1.14·37-s − 0.207·39-s − 0.156·41-s − 0.515·43-s + 0.491·45-s + 0.176·47-s − 0.428·49-s − 0.571·51-s − 0.689·53-s − 0.349·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.666423366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.666423366\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| good | 5 | \( 1 - 3.29T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 0.786T + 11T^{2} \) |
| 13 | \( 1 + 1.29T + 13T^{2} \) |
| 17 | \( 1 + 4.08T + 17T^{2} \) |
| 19 | \( 1 - 6.87T + 19T^{2} \) |
| 23 | \( 1 - 8.16T + 23T^{2} \) |
| 29 | \( 1 + 5.80T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 6.95T + 37T^{2} \) |
| 43 | \( 1 + 3.38T + 43T^{2} \) |
| 47 | \( 1 - 1.21T + 47T^{2} \) |
| 53 | \( 1 + 5.02T + 53T^{2} \) |
| 59 | \( 1 - 0.276T + 59T^{2} \) |
| 61 | \( 1 - 9.38T + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 + 4.08T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 + 2.27T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−9.694723796092624915523274081836, −9.323653254644331820306504888418, −8.505636112419638407874679533395, −7.40198322226929670169053893605, −6.75296066744715014330616741792, −5.34823066891702562800336500674, −5.10711305357400208120590027132, −3.51906581522189709470517657152, −2.35756533261555713327501586064, −1.52136416412664958303856438623,
1.52136416412664958303856438623, 2.35756533261555713327501586064, 3.51906581522189709470517657152, 5.10711305357400208120590027132, 5.34823066891702562800336500674, 6.75296066744715014330616741792, 7.40198322226929670169053893605, 8.505636112419638407874679533395, 9.323653254644331820306504888418, 9.694723796092624915523274081836
