L(s) = 1 | + 5.04·2-s + 3·3-s + 17.4·4-s + 20.1·5-s + 15.1·6-s − 15.4·7-s + 47.7·8-s + 9·9-s + 101.·10-s − 26.9·11-s + 52.3·12-s − 77.8·14-s + 60.3·15-s + 101.·16-s + 23.2·17-s + 45.4·18-s + 45.0·19-s + 351.·20-s − 46.2·21-s − 135.·22-s − 142.·23-s + 143.·24-s + 279.·25-s + 27·27-s − 269.·28-s + 2.29·29-s + 304.·30-s + ⋯ |
L(s) = 1 | + 1.78·2-s + 0.577·3-s + 2.18·4-s + 1.79·5-s + 1.02·6-s − 0.833·7-s + 2.10·8-s + 0.333·9-s + 3.20·10-s − 0.738·11-s + 1.25·12-s − 1.48·14-s + 1.03·15-s + 1.57·16-s + 0.331·17-s + 0.594·18-s + 0.544·19-s + 3.92·20-s − 0.480·21-s − 1.31·22-s − 1.28·23-s + 1.21·24-s + 2.23·25-s + 0.192·27-s − 1.81·28-s + 0.0146·29-s + 1.85·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(8.495852496\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.495852496\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 5.04T + 8T^{2} \) |
| 5 | \( 1 - 20.1T + 125T^{2} \) |
| 7 | \( 1 + 15.4T + 343T^{2} \) |
| 11 | \( 1 + 26.9T + 1.33e3T^{2} \) |
| 17 | \( 1 - 23.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 45.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 2.29T + 2.43e4T^{2} \) |
| 31 | \( 1 - 37.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 313.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 5.86T + 6.89e4T^{2} \) |
| 43 | \( 1 + 360.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 209.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 276.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 543.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 205.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 492.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 826.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 66.1T + 3.89e5T^{2} \) |
| 79 | \( 1 - 317.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 141.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 641.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.11e3T + 9.12e5T^{2} \) |
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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
−10.24724341459515405007781222623, −10.05997980635758338241303668052, −8.806114377591542763549404171012, −7.32902251706597652388178355682, −6.39318554517917693772032888313, −5.72162481710019673396901894975, −4.95738575003191919319154799520, −3.54996448776933078645185349850, −2.68412807772657100564231368243, −1.82798246467319153679739278870,
1.82798246467319153679739278870, 2.68412807772657100564231368243, 3.54996448776933078645185349850, 4.95738575003191919319154799520, 5.72162481710019673396901894975, 6.39318554517917693772032888313, 7.32902251706597652388178355682, 8.806114377591542763549404171012, 10.05997980635758338241303668052, 10.24724341459515405007781222623