| L(s) = 1 | + 1.51·3-s + 3.56·5-s − 1.45·7-s − 0.696·9-s + 5.38·11-s − 4.09·13-s + 5.41·15-s + 2.35·17-s − 3.02·19-s − 2.20·21-s − 1.50·23-s + 7.74·25-s − 5.61·27-s + 4.18·29-s + 10.5·31-s + 8.16·33-s − 5.18·35-s − 4.57·37-s − 6.20·39-s + 8.87·41-s + 9.31·43-s − 2.48·45-s + 3.06·47-s − 4.89·49-s + 3.57·51-s − 0.415·53-s + 19.2·55-s + ⋯ |
| L(s) = 1 | + 0.876·3-s + 1.59·5-s − 0.548·7-s − 0.232·9-s + 1.62·11-s − 1.13·13-s + 1.39·15-s + 0.570·17-s − 0.694·19-s − 0.480·21-s − 0.313·23-s + 1.54·25-s − 1.07·27-s + 0.776·29-s + 1.90·31-s + 1.42·33-s − 0.875·35-s − 0.752·37-s − 0.994·39-s + 1.38·41-s + 1.41·43-s − 0.370·45-s + 0.447·47-s − 0.698·49-s + 0.500·51-s − 0.0571·53-s + 2.59·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.538528734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.538528734\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 1.51T + 3T^{2} \) |
| 5 | \( 1 - 3.56T + 5T^{2} \) |
| 7 | \( 1 + 1.45T + 7T^{2} \) |
| 11 | \( 1 - 5.38T + 11T^{2} \) |
| 13 | \( 1 + 4.09T + 13T^{2} \) |
| 17 | \( 1 - 2.35T + 17T^{2} \) |
| 19 | \( 1 + 3.02T + 19T^{2} \) |
| 23 | \( 1 + 1.50T + 23T^{2} \) |
| 29 | \( 1 - 4.18T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 + 4.57T + 37T^{2} \) |
| 41 | \( 1 - 8.87T + 41T^{2} \) |
| 43 | \( 1 - 9.31T + 43T^{2} \) |
| 47 | \( 1 - 3.06T + 47T^{2} \) |
| 53 | \( 1 + 0.415T + 53T^{2} \) |
| 59 | \( 1 - 8.42T + 59T^{2} \) |
| 61 | \( 1 + 2.79T + 61T^{2} \) |
| 67 | \( 1 - 5.18T + 67T^{2} \) |
| 71 | \( 1 + 4.02T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 4.59T + 79T^{2} \) |
| 83 | \( 1 - 8.22T + 83T^{2} \) |
| 89 | \( 1 - 1.36T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−8.642967035488800537458875989044, −7.79782480595308594373116326605, −6.75489577165522435280727790569, −6.30911361637933009950146429550, −5.63251982087545249377676447833, −4.62024138270494653552512699095, −3.69979708889210990848187233794, −2.67074438486742892857277162889, −2.24359749457975604960724246546, −1.08079122309777195619415811915,
1.08079122309777195619415811915, 2.24359749457975604960724246546, 2.67074438486742892857277162889, 3.69979708889210990848187233794, 4.62024138270494653552512699095, 5.63251982087545249377676447833, 6.30911361637933009950146429550, 6.75489577165522435280727790569, 7.79782480595308594373116326605, 8.642967035488800537458875989044
