| L(s) = 1 | + 1.44·3-s − 3.20·5-s + 7-s − 0.914·9-s − 11-s − 13-s − 4.62·15-s − 2.31·17-s + 6.29·19-s + 1.44·21-s − 3.44·23-s + 5.26·25-s − 5.65·27-s + 7.75·29-s + 7.59·31-s − 1.44·33-s − 3.20·35-s + 4.91·37-s − 1.44·39-s − 6.73·41-s + 9.09·43-s + 2.92·45-s + 1.34·47-s + 49-s − 3.34·51-s − 7.22·53-s + 3.20·55-s + ⋯ |
| L(s) = 1 | + 0.833·3-s − 1.43·5-s + 0.377·7-s − 0.304·9-s − 0.301·11-s − 0.277·13-s − 1.19·15-s − 0.562·17-s + 1.44·19-s + 0.315·21-s − 0.717·23-s + 1.05·25-s − 1.08·27-s + 1.43·29-s + 1.36·31-s − 0.251·33-s − 0.541·35-s + 0.808·37-s − 0.231·39-s − 1.05·41-s + 1.38·43-s + 0.436·45-s + 0.196·47-s + 0.142·49-s − 0.469·51-s − 0.993·53-s + 0.432·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 1.44T + 3T^{2} \) |
| 5 | \( 1 + 3.20T + 5T^{2} \) |
| 17 | \( 1 + 2.31T + 17T^{2} \) |
| 19 | \( 1 - 6.29T + 19T^{2} \) |
| 23 | \( 1 + 3.44T + 23T^{2} \) |
| 29 | \( 1 - 7.75T + 29T^{2} \) |
| 31 | \( 1 - 7.59T + 31T^{2} \) |
| 37 | \( 1 - 4.91T + 37T^{2} \) |
| 41 | \( 1 + 6.73T + 41T^{2} \) |
| 43 | \( 1 - 9.09T + 43T^{2} \) |
| 47 | \( 1 - 1.34T + 47T^{2} \) |
| 53 | \( 1 + 7.22T + 53T^{2} \) |
| 59 | \( 1 + 4.77T + 59T^{2} \) |
| 61 | \( 1 + 1.83T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 + 3.88T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 0.193T + 79T^{2} \) |
| 83 | \( 1 + 0.916T + 83T^{2} \) |
| 89 | \( 1 + 9.66T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−7.71995071335206731945363577707, −7.07576657272667100614003280681, −6.17994563016900273433341619585, −5.22974740822898632051674728266, −4.47376829359897020304670830447, −3.91203941538843066568296367646, −2.97250238716923870836801033996, −2.60485075956949029185281494708, −1.21033659470009991605700608862, 0,
1.21033659470009991605700608862, 2.60485075956949029185281494708, 2.97250238716923870836801033996, 3.91203941538843066568296367646, 4.47376829359897020304670830447, 5.22974740822898632051674728266, 6.17994563016900273433341619585, 7.07576657272667100614003280681, 7.71995071335206731945363577707
