| L(s) = 1 | + 1.61·2-s − 2.23·3-s + 0.618·4-s + 5-s − 3.61·6-s − 0.236·7-s − 2.23·8-s + 2.00·9-s + 1.61·10-s + 4.23·11-s − 1.38·12-s − 0.381·14-s − 2.23·15-s − 4.85·16-s + 5.47·17-s + 3.23·18-s − 0.236·19-s + 0.618·20-s + 0.527·21-s + 6.85·22-s + 8.23·23-s + 5.00·24-s + 25-s + 2.23·27-s − 0.145·28-s + 1.47·29-s − 3.61·30-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 1.29·3-s + 0.309·4-s + 0.447·5-s − 1.47·6-s − 0.0892·7-s − 0.790·8-s + 0.666·9-s + 0.511·10-s + 1.27·11-s − 0.398·12-s − 0.102·14-s − 0.577·15-s − 1.21·16-s + 1.32·17-s + 0.762·18-s − 0.0541·19-s + 0.138·20-s + 0.115·21-s + 1.46·22-s + 1.71·23-s + 1.02·24-s + 0.200·25-s + 0.430·27-s − 0.0275·28-s + 0.273·29-s − 0.660·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.863388965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.863388965\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 3 | \( 1 + 2.23T + 3T^{2} \) |
| 7 | \( 1 + 0.236T + 7T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 17 | \( 1 - 5.47T + 17T^{2} \) |
| 19 | \( 1 + 0.236T + 19T^{2} \) |
| 23 | \( 1 - 8.23T + 23T^{2} \) |
| 29 | \( 1 - 1.47T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + 5.94T + 41T^{2} \) |
| 43 | \( 1 + 1.76T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 1.76T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 - 5.47T + 97T^{2} \) |
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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
−10.39574828657423261918685438574, −9.474170746869701467214880767006, −8.662245140735054791891351243610, −7.06000881323344409600106570694, −6.40264734229029026541490561412, −5.62138257008936027228317493943, −5.06398374962248635955954587260, −4.05905401375000477613221376889, −2.98475485981866324346373073696, −1.06298921105756188448608968146,
1.06298921105756188448608968146, 2.98475485981866324346373073696, 4.05905401375000477613221376889, 5.06398374962248635955954587260, 5.62138257008936027228317493943, 6.40264734229029026541490561412, 7.06000881323344409600106570694, 8.662245140735054791891351243610, 9.474170746869701467214880767006, 10.39574828657423261918685438574
