| L(s) = 1 | + 10.4·5-s − 6.42·7-s + 61.6·11-s − 64.8·13-s + 75.6·17-s + 10.3·19-s − 156.·23-s − 16.3·25-s − 53.7·29-s − 227.·31-s − 66.9·35-s − 10.3·37-s − 70.4·41-s − 298.·43-s − 89.9·47-s − 301.·49-s + 388.·53-s + 642.·55-s + 324·59-s − 324·61-s − 675.·65-s + 920.·67-s − 995.·71-s − 362.·73-s − 396.·77-s + 1.09e3·79-s − 791.·83-s + ⋯ |
| L(s) = 1 | + 0.932·5-s − 0.346·7-s + 1.69·11-s − 1.38·13-s + 1.07·17-s + 0.124·19-s − 1.42·23-s − 0.131·25-s − 0.344·29-s − 1.31·31-s − 0.323·35-s − 0.0458·37-s − 0.268·41-s − 1.05·43-s − 0.279·47-s − 0.879·49-s + 1.00·53-s + 1.57·55-s + 0.714·59-s − 0.680·61-s − 1.28·65-s + 1.67·67-s − 1.66·71-s − 0.580·73-s − 0.586·77-s + 1.56·79-s − 1.04·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 10.4T + 125T^{2} \) |
| 7 | \( 1 + 6.42T + 343T^{2} \) |
| 11 | \( 1 - 61.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 75.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 10.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 156.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 53.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 227.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 10.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 70.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 298.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 89.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 388.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 324T + 2.05e5T^{2} \) |
| 61 | \( 1 + 324T + 2.26e5T^{2} \) |
| 67 | \( 1 - 920.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 995.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 362.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.09e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 791.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 150.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.87e3T + 9.12e5T^{2} \) |
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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
−8.321340917760891867023696257557, −7.36815772976542513667876137099, −6.69471863740239650387205305697, −5.89522538231772232621084453372, −5.27604109642396013829914516997, −4.13813655066905111262332632306, −3.35615889906124741080397609818, −2.14659462555363910263270581179, −1.42575279509124588216066476077, 0,
1.42575279509124588216066476077, 2.14659462555363910263270581179, 3.35615889906124741080397609818, 4.13813655066905111262332632306, 5.27604109642396013829914516997, 5.89522538231772232621084453372, 6.69471863740239650387205305697, 7.36815772976542513667876137099, 8.321340917760891867023696257557
