| L(s) = 1 | − 1.35·2-s + 0.129·3-s − 0.155·4-s − 5-s − 0.175·6-s − 0.681·7-s + 2.92·8-s − 2.98·9-s + 1.35·10-s + 11-s − 0.0201·12-s + 4.22·13-s + 0.925·14-s − 0.129·15-s − 3.66·16-s − 7.92·17-s + 4.05·18-s + 0.754·19-s + 0.155·20-s − 0.0880·21-s − 1.35·22-s + 6.98·23-s + 0.378·24-s + 25-s − 5.74·26-s − 0.773·27-s + 0.105·28-s + ⋯ |
| L(s) = 1 | − 0.960·2-s + 0.0746·3-s − 0.0777·4-s − 0.447·5-s − 0.0716·6-s − 0.257·7-s + 1.03·8-s − 0.994·9-s + 0.429·10-s + 0.301·11-s − 0.00580·12-s + 1.17·13-s + 0.247·14-s − 0.0333·15-s − 0.916·16-s − 1.92·17-s + 0.954·18-s + 0.173·19-s + 0.0347·20-s − 0.0192·21-s − 0.289·22-s + 1.45·23-s + 0.0772·24-s + 0.200·25-s − 1.12·26-s − 0.148·27-s + 0.0200·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6083387827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6083387827\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 1.35T + 2T^{2} \) |
| 3 | \( 1 - 0.129T + 3T^{2} \) |
| 7 | \( 1 + 0.681T + 7T^{2} \) |
| 13 | \( 1 - 4.22T + 13T^{2} \) |
| 17 | \( 1 + 7.92T + 17T^{2} \) |
| 19 | \( 1 - 0.754T + 19T^{2} \) |
| 23 | \( 1 - 6.98T + 23T^{2} \) |
| 29 | \( 1 + 1.92T + 29T^{2} \) |
| 31 | \( 1 - 0.710T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 8.35T + 41T^{2} \) |
| 43 | \( 1 - 4.14T + 43T^{2} \) |
| 47 | \( 1 + 3.74T + 47T^{2} \) |
| 53 | \( 1 + 7.46T + 53T^{2} \) |
| 59 | \( 1 - 9.43T + 59T^{2} \) |
| 61 | \( 1 + 7.16T + 61T^{2} \) |
| 67 | \( 1 - 5.84T + 67T^{2} \) |
| 71 | \( 1 - 5.73T + 71T^{2} \) |
| 79 | \( 1 + 5.99T + 79T^{2} \) |
| 83 | \( 1 + 4.36T + 83T^{2} \) |
| 89 | \( 1 - 8.28T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−8.567090567034285805249041800806, −8.040800171042425595950166879082, −7.01819045337038612169179271500, −6.53649411125060445272447802663, −5.49223354749941687722472808576, −4.60804025281380726483696440707, −3.81716108566552097716830097431, −2.90682099825674043097277311396, −1.70663211770208302469835057461, −0.52083156560501838543170968971,
0.52083156560501838543170968971, 1.70663211770208302469835057461, 2.90682099825674043097277311396, 3.81716108566552097716830097431, 4.60804025281380726483696440707, 5.49223354749941687722472808576, 6.53649411125060445272447802663, 7.01819045337038612169179271500, 8.040800171042425595950166879082, 8.567090567034285805249041800806
