| L(s) = 1 | + 2·2-s + 4·4-s − 14.9·5-s + 21.7·7-s + 8·8-s − 29.8·10-s − 44.0·13-s + 43.5·14-s + 16·16-s + 24.9·17-s + 21.9·19-s − 59.6·20-s − 177.·23-s + 97.5·25-s − 88.0·26-s + 87.0·28-s + 149.·29-s + 75.1·31-s + 32·32-s + 49.8·34-s − 324.·35-s + 222.·37-s + 43.9·38-s − 119.·40-s + 253.·41-s + 130.·43-s − 355.·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.33·5-s + 1.17·7-s + 0.353·8-s − 0.943·10-s − 0.939·13-s + 0.831·14-s + 0.250·16-s + 0.355·17-s + 0.265·19-s − 0.667·20-s − 1.61·23-s + 0.780·25-s − 0.664·26-s + 0.587·28-s + 0.956·29-s + 0.435·31-s + 0.176·32-s + 0.251·34-s − 1.56·35-s + 0.987·37-s + 0.187·38-s − 0.471·40-s + 0.964·41-s + 0.463·43-s − 1.13·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.832841992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.832841992\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 14.9T + 125T^{2} \) |
| 7 | \( 1 - 21.7T + 343T^{2} \) |
| 13 | \( 1 + 44.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 24.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 21.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 177.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 149.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 75.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 222.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 253.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 130.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 499.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 12.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + 35.5T + 2.05e5T^{2} \) |
| 61 | \( 1 - 538.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 519.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 78.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.14e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 772.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 537.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 667.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 179.T + 9.12e5T^{2} \) |
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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
−8.188746370198027644770456621404, −7.996981512930361842073012018676, −7.31566792037329774135958493081, −6.32894165492962079317377370425, −5.30394246791144273996480985154, −4.54260801875513468905736380601, −4.06436613325523943826158494798, −2.99941819830765856385189808683, −1.97231878644996296820815392975, −0.67953142614507858342038550186,
0.67953142614507858342038550186, 1.97231878644996296820815392975, 2.99941819830765856385189808683, 4.06436613325523943826158494798, 4.54260801875513468905736380601, 5.30394246791144273996480985154, 6.32894165492962079317377370425, 7.31566792037329774135958493081, 7.996981512930361842073012018676, 8.188746370198027644770456621404
