| L(s) = 1 | − 7-s + 11-s − 3·17-s + 19-s + 2·23-s − 5·25-s − 7·29-s + 8·31-s + 8·37-s − 3·41-s − 4·43-s − 7·47-s + 49-s + 11·53-s − 14·59-s + 7·61-s − 2·67-s + 2·71-s + 4·73-s − 77-s − 79-s − 6·83-s + 15·89-s − 16·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.301·11-s − 0.727·17-s + 0.229·19-s + 0.417·23-s − 25-s − 1.29·29-s + 1.43·31-s + 1.31·37-s − 0.468·41-s − 0.609·43-s − 1.02·47-s + 1/7·49-s + 1.51·53-s − 1.82·59-s + 0.896·61-s − 0.244·67-s + 0.237·71-s + 0.468·73-s − 0.113·77-s − 0.112·79-s − 0.658·83-s + 1.58·89-s − 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.567425835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.567425835\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.75449874180185, −13.33386027109608, −13.18788594230123, −12.39927247410324, −11.90659592382778, −11.47659267921446, −11.04543232914403, −10.40501916863585, −9.869333488903724, −9.414658101627516, −9.053011570771350, −8.264585626747330, −7.961085348263203, −7.258414092190902, −6.722399088430432, −6.285265493080938, −5.707930870155268, −5.118548922077947, −4.426484108021588, −3.974002214959772, −3.308297886706747, −2.684208483450791, −2.023498132489467, −1.297252858487134, −0.4115533983182896,
0.4115533983182896, 1.297252858487134, 2.023498132489467, 2.684208483450791, 3.308297886706747, 3.974002214959772, 4.426484108021588, 5.118548922077947, 5.707930870155268, 6.285265493080938, 6.722399088430432, 7.258414092190902, 7.961085348263203, 8.264585626747330, 9.053011570771350, 9.414658101627516, 9.869333488903724, 10.40501916863585, 11.04543232914403, 11.47659267921446, 11.90659592382778, 12.39927247410324, 13.18788594230123, 13.33386027109608, 13.75449874180185
