| L(s) = 1 | − 2.16·2-s + 2.67·4-s − 3.42·5-s + 1.52·7-s − 1.46·8-s + 7.40·10-s + 11-s + 3.58·13-s − 3.30·14-s − 2.18·16-s − 5.94·17-s − 1.65·19-s − 9.17·20-s − 2.16·22-s + 3.88·23-s + 6.73·25-s − 7.75·26-s + 4.08·28-s + 9.39·29-s − 5.87·31-s + 7.65·32-s + 12.8·34-s − 5.22·35-s + 3.29·37-s + 3.58·38-s + 5.02·40-s − 11.2·41-s + ⋯ |
| L(s) = 1 | − 1.52·2-s + 1.33·4-s − 1.53·5-s + 0.576·7-s − 0.518·8-s + 2.34·10-s + 0.301·11-s + 0.994·13-s − 0.881·14-s − 0.545·16-s − 1.44·17-s − 0.380·19-s − 2.05·20-s − 0.461·22-s + 0.810·23-s + 1.34·25-s − 1.52·26-s + 0.772·28-s + 1.74·29-s − 1.05·31-s + 1.35·32-s + 2.20·34-s − 0.883·35-s + 0.541·37-s + 0.581·38-s + 0.795·40-s − 1.75·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5614142888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5614142888\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 2.16T + 2T^{2} \) |
| 5 | \( 1 + 3.42T + 5T^{2} \) |
| 7 | \( 1 - 1.52T + 7T^{2} \) |
| 13 | \( 1 - 3.58T + 13T^{2} \) |
| 17 | \( 1 + 5.94T + 17T^{2} \) |
| 19 | \( 1 + 1.65T + 19T^{2} \) |
| 23 | \( 1 - 3.88T + 23T^{2} \) |
| 29 | \( 1 - 9.39T + 29T^{2} \) |
| 31 | \( 1 + 5.87T + 31T^{2} \) |
| 37 | \( 1 - 3.29T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 4.96T + 43T^{2} \) |
| 47 | \( 1 + 0.704T + 47T^{2} \) |
| 53 | \( 1 - 5.18T + 53T^{2} \) |
| 59 | \( 1 - 2.75T + 59T^{2} \) |
| 67 | \( 1 - 7.28T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 2.44T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 0.417T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−8.240702585813879762817389385542, −7.66322093931029759832877376596, −6.80352085314997634677289920653, −6.54432281470688229418685859688, −5.06054896929425006209622092824, −4.33868042267274923692673648231, −3.66077601071146449591158596431, −2.52359297094748644779245475586, −1.44565274993617887405513217022, −0.53365093071450365134683378522,
0.53365093071450365134683378522, 1.44565274993617887405513217022, 2.52359297094748644779245475586, 3.66077601071146449591158596431, 4.33868042267274923692673648231, 5.06054896929425006209622092824, 6.54432281470688229418685859688, 6.80352085314997634677289920653, 7.66322093931029759832877376596, 8.240702585813879762817389385542
