| L(s) = 1 | − 0.111·5-s − 3.78·7-s + 5.64·11-s + 6.56·13-s + 2.37·17-s + 4.46·19-s + 2.90·23-s − 4.98·25-s + 0.383·29-s − 5.50·31-s + 0.422·35-s + 6.40·37-s − 4.20·41-s + 3.54·43-s + 0.255·47-s + 7.32·49-s − 12.1·53-s − 0.630·55-s − 9.65·59-s − 2.24·61-s − 0.732·65-s + 10.6·67-s + 6.99·71-s + 1.95·73-s − 21.3·77-s + 14.0·79-s − 2.74·83-s + ⋯ |
| L(s) = 1 | − 0.0499·5-s − 1.43·7-s + 1.70·11-s + 1.81·13-s + 0.577·17-s + 1.02·19-s + 0.604·23-s − 0.997·25-s + 0.0712·29-s − 0.987·31-s + 0.0714·35-s + 1.05·37-s − 0.657·41-s + 0.541·43-s + 0.0372·47-s + 1.04·49-s − 1.67·53-s − 0.0849·55-s − 1.25·59-s − 0.288·61-s − 0.0908·65-s + 1.30·67-s + 0.829·71-s + 0.228·73-s − 2.43·77-s + 1.58·79-s − 0.301·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.119155846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.119155846\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 0.111T + 5T^{2} \) |
| 7 | \( 1 + 3.78T + 7T^{2} \) |
| 11 | \( 1 - 5.64T + 11T^{2} \) |
| 13 | \( 1 - 6.56T + 13T^{2} \) |
| 17 | \( 1 - 2.37T + 17T^{2} \) |
| 19 | \( 1 - 4.46T + 19T^{2} \) |
| 23 | \( 1 - 2.90T + 23T^{2} \) |
| 29 | \( 1 - 0.383T + 29T^{2} \) |
| 31 | \( 1 + 5.50T + 31T^{2} \) |
| 37 | \( 1 - 6.40T + 37T^{2} \) |
| 41 | \( 1 + 4.20T + 41T^{2} \) |
| 43 | \( 1 - 3.54T + 43T^{2} \) |
| 47 | \( 1 - 0.255T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 9.65T + 59T^{2} \) |
| 61 | \( 1 + 2.24T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 6.99T + 71T^{2} \) |
| 73 | \( 1 - 1.95T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 + 2.74T + 83T^{2} \) |
| 89 | \( 1 + 9.44T + 89T^{2} \) |
| 97 | \( 1 + 6.83T + 97T^{2} \) |
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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
−8.087529949395195212181402648698, −7.28282734944658699176367232670, −6.37756086874552837946239955148, −6.26229973471381045487453593905, −5.39949019926765079436394225641, −4.14535002523516614805803377507, −3.54576373968763018840996959525, −3.15900688650961563665352222646, −1.64512870007334456560401244242, −0.820788894604101841275666140544,
0.820788894604101841275666140544, 1.64512870007334456560401244242, 3.15900688650961563665352222646, 3.54576373968763018840996959525, 4.14535002523516614805803377507, 5.39949019926765079436394225641, 6.26229973471381045487453593905, 6.37756086874552837946239955148, 7.28282734944658699176367232670, 8.087529949395195212181402648698
