| L(s) = 1 | − 0.741·2-s − 1.44·4-s − 3.30·5-s + 1.44·7-s + 2.55·8-s + 2.44·10-s + 1.81·11-s + 13-s − 1.07·14-s + 1.00·16-s + 6.60·17-s + 4.78·20-s − 1.34·22-s − 4.78·23-s + 5.89·25-s − 0.741·26-s − 2.10·28-s − 9.57·29-s + 4.55·31-s − 5.86·32-s − 4.89·34-s − 4.78·35-s + 5.89·37-s − 8.44·40-s − 2.96·41-s − 8.34·43-s − 2.63·44-s + ⋯ |
| L(s) = 1 | − 0.524·2-s − 0.724·4-s − 1.47·5-s + 0.547·7-s + 0.904·8-s + 0.774·10-s + 0.547·11-s + 0.277·13-s − 0.287·14-s + 0.250·16-s + 1.60·17-s + 1.07·20-s − 0.287·22-s − 0.997·23-s + 1.17·25-s − 0.145·26-s − 0.397·28-s − 1.77·29-s + 0.817·31-s − 1.03·32-s − 0.840·34-s − 0.808·35-s + 0.969·37-s − 1.33·40-s − 0.463·41-s − 1.27·43-s − 0.397·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8114075243\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8114075243\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.741T + 2T^{2} \) |
| 5 | \( 1 + 3.30T + 5T^{2} \) |
| 7 | \( 1 - 1.44T + 7T^{2} \) |
| 11 | \( 1 - 1.81T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 6.60T + 17T^{2} \) |
| 23 | \( 1 + 4.78T + 23T^{2} \) |
| 29 | \( 1 + 9.57T + 29T^{2} \) |
| 31 | \( 1 - 4.55T + 31T^{2} \) |
| 37 | \( 1 - 5.89T + 37T^{2} \) |
| 41 | \( 1 + 2.96T + 41T^{2} \) |
| 43 | \( 1 + 8.34T + 43T^{2} \) |
| 47 | \( 1 + 2.96T + 47T^{2} \) |
| 53 | \( 1 + 3.30T + 53T^{2} \) |
| 59 | \( 1 + 8.42T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 - 9.57T + 71T^{2} \) |
| 73 | \( 1 - 5T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 + 3.63T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.334296885532885557619226161923, −8.022461046716400856997926302319, −7.61884064847101875166303178430, −6.57305587889787172224618017606, −5.44790387524606520473118604910, −4.70659832773773087372875652171, −3.83928013179181337966596164506, −3.45592840940633254991180364567, −1.67896800849665446900339700592, −0.61786061552046168591442608637,
0.61786061552046168591442608637, 1.67896800849665446900339700592, 3.45592840940633254991180364567, 3.83928013179181337966596164506, 4.70659832773773087372875652171, 5.44790387524606520473118604910, 6.57305587889787172224618017606, 7.61884064847101875166303178430, 8.022461046716400856997926302319, 8.334296885532885557619226161923
