| L(s) = 1 | + 1.11i·5-s + 1.19i·7-s + 3.72·11-s + 0.113·13-s − 2.89i·17-s − i·19-s + 1.35·23-s + 3.76·25-s − 0.860i·29-s − 9.01i·31-s − 1.32·35-s + 5.64·37-s + 10.5i·41-s − 2.47i·43-s − 5.05·47-s + ⋯ |
| L(s) = 1 | + 0.496i·5-s + 0.451i·7-s + 1.12·11-s + 0.0314·13-s − 0.701i·17-s − 0.229i·19-s + 0.282·23-s + 0.753·25-s − 0.159i·29-s − 1.61i·31-s − 0.224·35-s + 0.927·37-s + 1.64i·41-s − 0.376i·43-s − 0.737·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.005939331\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.005939331\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + iT \) |
| good | 5 | \( 1 - 1.11iT - 5T^{2} \) |
| 7 | \( 1 - 1.19iT - 7T^{2} \) |
| 11 | \( 1 - 3.72T + 11T^{2} \) |
| 13 | \( 1 - 0.113T + 13T^{2} \) |
| 17 | \( 1 + 2.89iT - 17T^{2} \) |
| 23 | \( 1 - 1.35T + 23T^{2} \) |
| 29 | \( 1 + 0.860iT - 29T^{2} \) |
| 31 | \( 1 + 9.01iT - 31T^{2} \) |
| 37 | \( 1 - 5.64T + 37T^{2} \) |
| 41 | \( 1 - 10.5iT - 41T^{2} \) |
| 43 | \( 1 + 2.47iT - 43T^{2} \) |
| 47 | \( 1 + 5.05T + 47T^{2} \) |
| 53 | \( 1 - 9.51iT - 53T^{2} \) |
| 59 | \( 1 - 8.73T + 59T^{2} \) |
| 61 | \( 1 + 8.52T + 61T^{2} \) |
| 67 | \( 1 + 0.857iT - 67T^{2} \) |
| 71 | \( 1 + 1.91T + 71T^{2} \) |
| 73 | \( 1 - 3.06T + 73T^{2} \) |
| 79 | \( 1 + 7.62iT - 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 13.1iT - 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−9.106684546915853540780606476959, −8.075359181661862681439157239699, −7.33491586662587085891078791443, −6.51394665682912242642257653535, −5.99621326827395658344063933644, −4.92123300130349441300388325487, −4.12466995254342940401351392188, −3.10835885180261416396066778141, −2.30750173144674365471171911672, −0.962037633416475640624917878995,
0.889678226430914438320526627036, 1.81716577732818575333940141232, 3.23838830473668401743999898311, 4.00657385309672273792848407889, 4.79486694285351332083355310093, 5.67208038289866892274724023510, 6.61375455193235354779149364576, 7.11743135902885977037476890475, 8.169756617226092554526723122017, 8.778114488094646291513999115625
