| L(s) = 1 | + 2-s + 3.12·3-s + 4-s − 3.62·5-s + 3.12·6-s + 7-s + 8-s + 6.79·9-s − 3.62·10-s + 5.44·11-s + 3.12·12-s − 0.710·13-s + 14-s − 11.3·15-s + 16-s + 0.171·17-s + 6.79·18-s + 0.530·19-s − 3.62·20-s + 3.12·21-s + 5.44·22-s + 1.99·23-s + 3.12·24-s + 8.15·25-s − 0.710·26-s + 11.8·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.80·3-s + 0.5·4-s − 1.62·5-s + 1.27·6-s + 0.377·7-s + 0.353·8-s + 2.26·9-s − 1.14·10-s + 1.64·11-s + 0.903·12-s − 0.196·13-s + 0.267·14-s − 2.93·15-s + 0.250·16-s + 0.0416·17-s + 1.60·18-s + 0.121·19-s − 0.811·20-s + 0.682·21-s + 1.16·22-s + 0.417·23-s + 0.638·24-s + 1.63·25-s − 0.139·26-s + 2.28·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.745401600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.745401600\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 - 3.12T + 3T^{2} \) |
| 5 | \( 1 + 3.62T + 5T^{2} \) |
| 11 | \( 1 - 5.44T + 11T^{2} \) |
| 13 | \( 1 + 0.710T + 13T^{2} \) |
| 17 | \( 1 - 0.171T + 17T^{2} \) |
| 19 | \( 1 - 0.530T + 19T^{2} \) |
| 23 | \( 1 - 1.99T + 23T^{2} \) |
| 29 | \( 1 + 4.18T + 29T^{2} \) |
| 31 | \( 1 + 6.72T + 31T^{2} \) |
| 37 | \( 1 - 6.99T + 37T^{2} \) |
| 41 | \( 1 - 8.95T + 41T^{2} \) |
| 43 | \( 1 + 0.968T + 43T^{2} \) |
| 47 | \( 1 - 5.77T + 47T^{2} \) |
| 53 | \( 1 + 6.96T + 53T^{2} \) |
| 59 | \( 1 + 8.18T + 59T^{2} \) |
| 61 | \( 1 - 4.37T + 61T^{2} \) |
| 67 | \( 1 + 7.47T + 67T^{2} \) |
| 71 | \( 1 + 0.270T + 71T^{2} \) |
| 73 | \( 1 - 0.0463T + 73T^{2} \) |
| 79 | \( 1 - 2.05T + 79T^{2} \) |
| 83 | \( 1 + 5.02T + 83T^{2} \) |
| 89 | \( 1 + 1.65T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−7.88353256164902432381444775101, −7.47506692050017253183582883169, −7.05116601764084656052474160373, −6.02369251291859010900107822068, −4.64836589808893975907244297293, −4.21087104687114049842751289474, −3.63518472146425768195566475657, −3.13774046236730824231283676900, −2.10849228676205167673900419415, −1.12795401364864219505082279812,
1.12795401364864219505082279812, 2.10849228676205167673900419415, 3.13774046236730824231283676900, 3.63518472146425768195566475657, 4.21087104687114049842751289474, 4.64836589808893975907244297293, 6.02369251291859010900107822068, 7.05116601764084656052474160373, 7.47506692050017253183582883169, 7.88353256164902432381444775101
