| L(s) = 1 | − 17.9·5-s + 7.28·7-s + 4.61·11-s + 29.8·13-s + 67.9·17-s + 111.·19-s − 218.·23-s + 198.·25-s + 34.2·29-s − 77.7·31-s − 130.·35-s − 347.·37-s + 234.·41-s + 53.3·43-s − 385.·47-s − 289.·49-s − 461.·53-s − 82.9·55-s + 7.16·59-s + 416.·61-s − 537.·65-s − 869.·67-s − 585.·71-s − 733.·73-s + 33.6·77-s + 1.17e3·79-s − 67.4·83-s + ⋯ |
| L(s) = 1 | − 1.60·5-s + 0.393·7-s + 0.126·11-s + 0.637·13-s + 0.969·17-s + 1.34·19-s − 1.98·23-s + 1.58·25-s + 0.219·29-s − 0.450·31-s − 0.632·35-s − 1.54·37-s + 0.893·41-s + 0.189·43-s − 1.19·47-s − 0.845·49-s − 1.19·53-s − 0.203·55-s + 0.0158·59-s + 0.873·61-s − 1.02·65-s − 1.58·67-s − 0.979·71-s − 1.17·73-s + 0.0497·77-s + 1.66·79-s − 0.0891·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 17.9T + 125T^{2} \) |
| 7 | \( 1 - 7.28T + 343T^{2} \) |
| 11 | \( 1 - 4.61T + 1.33e3T^{2} \) |
| 13 | \( 1 - 29.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 67.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 111.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 218.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 34.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 77.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 347.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 234.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 53.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 385.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 461.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 7.16T + 2.05e5T^{2} \) |
| 61 | \( 1 - 416.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 869.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 585.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 733.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 67.4T + 5.71e5T^{2} \) |
| 89 | \( 1 + 965.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.43T + 9.12e5T^{2} \) |
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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
−9.768722801882447711419783941866, −8.617455269185724848676454506360, −7.87254327895731654860067572181, −7.40167098089854465601883757655, −6.11012215582931254705310708340, −4.98644993586293269556998520146, −3.91766970247283554593030108831, −3.25267429593388203232271509935, −1.40339986641973325446299452637, 0,
1.40339986641973325446299452637, 3.25267429593388203232271509935, 3.91766970247283554593030108831, 4.98644993586293269556998520146, 6.11012215582931254705310708340, 7.40167098089854465601883757655, 7.87254327895731654860067572181, 8.617455269185724848676454506360, 9.768722801882447711419783941866
