| L(s) = 1 | + 2-s − 0.381·3-s + 4-s + 3.85·5-s − 0.381·6-s + 7-s + 8-s − 2.85·9-s + 3.85·10-s + 2.38·11-s − 0.381·12-s − 0.763·13-s + 14-s − 1.47·15-s + 16-s − 6.47·17-s − 2.85·18-s + 3.85·20-s − 0.381·21-s + 2.38·22-s + 7.70·23-s − 0.381·24-s + 9.85·25-s − 0.763·26-s + 2.23·27-s + 28-s + 2.14·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.220·3-s + 0.5·4-s + 1.72·5-s − 0.155·6-s + 0.377·7-s + 0.353·8-s − 0.951·9-s + 1.21·10-s + 0.718·11-s − 0.110·12-s − 0.211·13-s + 0.267·14-s − 0.380·15-s + 0.250·16-s − 1.56·17-s − 0.672·18-s + 0.861·20-s − 0.0833·21-s + 0.507·22-s + 1.60·23-s − 0.0779·24-s + 1.97·25-s − 0.149·26-s + 0.430·27-s + 0.188·28-s + 0.398·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.204970554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.204970554\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.381T + 3T^{2} \) |
| 5 | \( 1 - 3.85T + 5T^{2} \) |
| 11 | \( 1 - 2.38T + 11T^{2} \) |
| 13 | \( 1 + 0.763T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 23 | \( 1 - 7.70T + 23T^{2} \) |
| 29 | \( 1 - 2.14T + 29T^{2} \) |
| 31 | \( 1 + 8.47T + 31T^{2} \) |
| 37 | \( 1 - 1.85T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 5.38T + 43T^{2} \) |
| 47 | \( 1 - 2.85T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 - 2.14T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 6.94T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 2.76T + 73T^{2} \) |
| 79 | \( 1 + 2.32T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 - 3.61T + 89T^{2} \) |
| 97 | \( 1 - 2.85T + 97T^{2} \) |
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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
−8.430918513188159061896563010300, −7.03883930306591710024813396994, −6.72305413844112609139115057050, −5.88481210995094417572137475359, −5.39751163031909061757464077215, −4.79071729405998268668481335376, −3.79420894479465020591641029311, −2.58475834148330168464255290766, −2.21483928580706306924034669323, −1.06267392337163156696937956249,
1.06267392337163156696937956249, 2.21483928580706306924034669323, 2.58475834148330168464255290766, 3.79420894479465020591641029311, 4.79071729405998268668481335376, 5.39751163031909061757464077215, 5.88481210995094417572137475359, 6.72305413844112609139115057050, 7.03883930306591710024813396994, 8.430918513188159061896563010300
