| L(s) = 1 | − 2-s + 2.34·3-s + 4-s + 1.32·5-s − 2.34·6-s + 3.84·7-s − 8-s + 2.51·9-s − 1.32·10-s + 2.09·11-s + 2.34·12-s + 0.110·13-s − 3.84·14-s + 3.11·15-s + 16-s + 5.18·17-s − 2.51·18-s + 19-s + 1.32·20-s + 9.03·21-s − 2.09·22-s + 1.27·23-s − 2.34·24-s − 3.23·25-s − 0.110·26-s − 1.13·27-s + 3.84·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.35·3-s + 0.5·4-s + 0.593·5-s − 0.958·6-s + 1.45·7-s − 0.353·8-s + 0.838·9-s − 0.420·10-s + 0.632·11-s + 0.677·12-s + 0.0305·13-s − 1.02·14-s + 0.805·15-s + 0.250·16-s + 1.25·17-s − 0.592·18-s + 0.229·19-s + 0.296·20-s + 1.97·21-s − 0.447·22-s + 0.265·23-s − 0.479·24-s − 0.647·25-s − 0.0215·26-s − 0.219·27-s + 0.727·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.797449811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.797449811\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 2.34T + 3T^{2} \) |
| 5 | \( 1 - 1.32T + 5T^{2} \) |
| 7 | \( 1 - 3.84T + 7T^{2} \) |
| 11 | \( 1 - 2.09T + 11T^{2} \) |
| 13 | \( 1 - 0.110T + 13T^{2} \) |
| 17 | \( 1 - 5.18T + 17T^{2} \) |
| 23 | \( 1 - 1.27T + 23T^{2} \) |
| 29 | \( 1 - 0.445T + 29T^{2} \) |
| 31 | \( 1 + 7.71T + 31T^{2} \) |
| 37 | \( 1 + 4.69T + 37T^{2} \) |
| 41 | \( 1 - 5.09T + 41T^{2} \) |
| 43 | \( 1 - 6.41T + 43T^{2} \) |
| 47 | \( 1 + 9.70T + 47T^{2} \) |
| 53 | \( 1 - 3.65T + 53T^{2} \) |
| 59 | \( 1 + 1.86T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 - 7.88T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 6.10T + 73T^{2} \) |
| 79 | \( 1 - 4.57T + 79T^{2} \) |
| 83 | \( 1 - 5.75T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 - 0.279T + 97T^{2} \) |
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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
−8.002453600574346092421328859065, −7.48952137467690026373033349644, −6.74649053218411111635513229349, −5.66534753581685644473861283144, −5.18723306824592566991221094941, −4.00284926470729608942571793605, −3.41938942313973678882221607878, −2.37370509014532795376790271978, −1.81981030478340566818468843835, −1.10317335153600621451577828244,
1.10317335153600621451577828244, 1.81981030478340566818468843835, 2.37370509014532795376790271978, 3.41938942313973678882221607878, 4.00284926470729608942571793605, 5.18723306824592566991221094941, 5.66534753581685644473861283144, 6.74649053218411111635513229349, 7.48952137467690026373033349644, 8.002453600574346092421328859065
