| L(s) = 1 | − 7.11·3-s + 16.3·5-s − 5.74·7-s + 23.6·9-s − 23.3·11-s − 18.0·13-s − 116.·15-s − 24.1·17-s + 12.0·19-s + 40.8·21-s + 144.·23-s + 143.·25-s + 24.1·27-s + 29·29-s − 6.27·31-s + 166.·33-s − 94.1·35-s + 28.6·37-s + 128.·39-s − 436.·41-s − 495.·43-s + 386.·45-s − 351.·47-s − 309.·49-s + 171.·51-s − 58.0·53-s − 382.·55-s + ⋯ |
| L(s) = 1 | − 1.36·3-s + 1.46·5-s − 0.310·7-s + 0.874·9-s − 0.640·11-s − 0.384·13-s − 2.00·15-s − 0.344·17-s + 0.145·19-s + 0.424·21-s + 1.31·23-s + 1.14·25-s + 0.171·27-s + 0.185·29-s − 0.0363·31-s + 0.877·33-s − 0.454·35-s + 0.127·37-s + 0.526·39-s − 1.66·41-s − 1.75·43-s + 1.28·45-s − 1.09·47-s − 0.903·49-s + 0.471·51-s − 0.150·53-s − 0.938·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 + 7.11T + 27T^{2} \) |
| 5 | \( 1 - 16.3T + 125T^{2} \) |
| 7 | \( 1 + 5.74T + 343T^{2} \) |
| 11 | \( 1 + 23.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 18.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 24.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 12.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 144.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 6.27T + 2.97e4T^{2} \) |
| 37 | \( 1 - 28.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 436.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 495.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 351.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 58.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 485.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 607.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 296.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 330.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 662.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 145.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.07e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 851.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 227.T + 9.12e5T^{2} \) |
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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
−10.17854398544000110333678351332, −9.668952505798922918735181509789, −8.483369455794518322761820063663, −6.92843770417628211187471766082, −6.37238927361438368630624636733, −5.35170809408444721761632422925, −4.92411086021305858145495010457, −2.93601064083945242965727030829, −1.53689782559079845939974372089, 0,
1.53689782559079845939974372089, 2.93601064083945242965727030829, 4.92411086021305858145495010457, 5.35170809408444721761632422925, 6.37238927361438368630624636733, 6.92843770417628211187471766082, 8.483369455794518322761820063663, 9.668952505798922918735181509789, 10.17854398544000110333678351332
