| L(s) = 1 | − 0.608·2-s + 3-s − 1.62·4-s − 0.608·6-s − 3.83·7-s + 2.20·8-s + 9-s − 0.466·11-s − 1.62·12-s + 1.86·13-s + 2.33·14-s + 1.91·16-s + 0.268·17-s − 0.608·18-s − 5.20·19-s − 3.83·21-s + 0.283·22-s − 0.943·23-s + 2.20·24-s − 1.13·26-s + 27-s + 6.25·28-s − 5.99·29-s + 3.89·31-s − 5.58·32-s − 0.466·33-s − 0.163·34-s + ⋯ |
| L(s) = 1 | − 0.430·2-s + 0.577·3-s − 0.814·4-s − 0.248·6-s − 1.45·7-s + 0.780·8-s + 0.333·9-s − 0.140·11-s − 0.470·12-s + 0.518·13-s + 0.624·14-s + 0.478·16-s + 0.0650·17-s − 0.143·18-s − 1.19·19-s − 0.837·21-s + 0.0605·22-s − 0.196·23-s + 0.450·24-s − 0.223·26-s + 0.192·27-s + 1.18·28-s − 1.11·29-s + 0.699·31-s − 0.986·32-s − 0.0812·33-s − 0.0279·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9386040507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9386040507\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 0.608T + 2T^{2} \) |
| 7 | \( 1 + 3.83T + 7T^{2} \) |
| 11 | \( 1 + 0.466T + 11T^{2} \) |
| 13 | \( 1 - 1.86T + 13T^{2} \) |
| 17 | \( 1 - 0.268T + 17T^{2} \) |
| 19 | \( 1 + 5.20T + 19T^{2} \) |
| 23 | \( 1 + 0.943T + 23T^{2} \) |
| 29 | \( 1 + 5.99T + 29T^{2} \) |
| 31 | \( 1 - 3.89T + 31T^{2} \) |
| 37 | \( 1 + 4.18T + 37T^{2} \) |
| 41 | \( 1 - 2.27T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 53 | \( 1 + 4.42T + 53T^{2} \) |
| 59 | \( 1 - 6.97T + 59T^{2} \) |
| 61 | \( 1 + 8.00T + 61T^{2} \) |
| 67 | \( 1 - 2.37T + 67T^{2} \) |
| 71 | \( 1 + 6.83T + 71T^{2} \) |
| 73 | \( 1 - 7.07T + 73T^{2} \) |
| 79 | \( 1 - 1.79T + 79T^{2} \) |
| 83 | \( 1 + 9.36T + 83T^{2} \) |
| 89 | \( 1 - 5.44T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−8.808704433431874437138820195149, −7.964858400100304516841593207299, −7.26231803740218411479478727140, −6.36093473353805170891571108849, −5.70347623385206040198015017215, −4.52202515192261393999578238245, −3.84337438105916651215961604834, −3.12771547947744154768919068020, −1.99670205628018385412299311690, −0.58251674099735174222063472715,
0.58251674099735174222063472715, 1.99670205628018385412299311690, 3.12771547947744154768919068020, 3.84337438105916651215961604834, 4.52202515192261393999578238245, 5.70347623385206040198015017215, 6.36093473353805170891571108849, 7.26231803740218411479478727140, 7.964858400100304516841593207299, 8.808704433431874437138820195149
