| L(s) = 1 | − 2-s − 1.52·3-s + 4-s − 0.0721·5-s + 1.52·6-s + 3.89·7-s − 8-s − 0.680·9-s + 0.0721·10-s + 1.26·11-s − 1.52·12-s + 3.78·13-s − 3.89·14-s + 0.109·15-s + 16-s − 7.52·17-s + 0.680·18-s + 2.59·19-s − 0.0721·20-s − 5.94·21-s − 1.26·22-s + 1.46·23-s + 1.52·24-s − 4.99·25-s − 3.78·26-s + 5.60·27-s + 3.89·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.879·3-s + 0.5·4-s − 0.0322·5-s + 0.621·6-s + 1.47·7-s − 0.353·8-s − 0.226·9-s + 0.0228·10-s + 0.381·11-s − 0.439·12-s + 1.05·13-s − 1.04·14-s + 0.0283·15-s + 0.250·16-s − 1.82·17-s + 0.160·18-s + 0.594·19-s − 0.0161·20-s − 1.29·21-s − 0.269·22-s + 0.304·23-s + 0.310·24-s − 0.998·25-s − 0.743·26-s + 1.07·27-s + 0.737·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.147145953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.147145953\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 + 1.52T + 3T^{2} \) |
| 5 | \( 1 + 0.0721T + 5T^{2} \) |
| 7 | \( 1 - 3.89T + 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 13 | \( 1 - 3.78T + 13T^{2} \) |
| 17 | \( 1 + 7.52T + 17T^{2} \) |
| 19 | \( 1 - 2.59T + 19T^{2} \) |
| 23 | \( 1 - 1.46T + 23T^{2} \) |
| 29 | \( 1 - 5.05T + 29T^{2} \) |
| 37 | \( 1 + 3.19T + 37T^{2} \) |
| 41 | \( 1 - 6.54T + 41T^{2} \) |
| 43 | \( 1 - 4.82T + 43T^{2} \) |
| 47 | \( 1 + 7.18T + 47T^{2} \) |
| 53 | \( 1 + 6.50T + 53T^{2} \) |
| 59 | \( 1 - 7.31T + 59T^{2} \) |
| 61 | \( 1 - 4.13T + 61T^{2} \) |
| 67 | \( 1 - 9.24T + 67T^{2} \) |
| 71 | \( 1 + 0.0717T + 71T^{2} \) |
| 73 | \( 1 + 6.45T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 - 0.0614T + 83T^{2} \) |
| 89 | \( 1 - 5.60T + 89T^{2} \) |
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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
−8.220833297544903722704681823222, −7.44574554160176787061203814554, −6.56938936671287757581782536640, −6.11649651385542041349723671504, −5.23881315665613522174755020047, −4.62728966037185297597840933017, −3.74177337024072794749883407993, −2.46552748815199711906146461693, −1.59719206847218224590600102089, −0.68871029627818796455354785563,
0.68871029627818796455354785563, 1.59719206847218224590600102089, 2.46552748815199711906146461693, 3.74177337024072794749883407993, 4.62728966037185297597840933017, 5.23881315665613522174755020047, 6.11649651385542041349723671504, 6.56938936671287757581782536640, 7.44574554160176787061203814554, 8.220833297544903722704681823222
