| L(s) = 1 | − 3-s + 2·5-s − 2·9-s + 4·11-s + 2·13-s − 2·15-s − 6·17-s − 19-s + 6·23-s − 25-s + 5·27-s + 4·29-s + 8·31-s − 4·33-s + 2·37-s − 2·39-s + 2·43-s − 4·45-s + 7·47-s − 7·49-s + 6·51-s + 9·53-s + 8·55-s + 57-s − 5·59-s − 2·61-s + 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 2/3·9-s + 1.20·11-s + 0.554·13-s − 0.516·15-s − 1.45·17-s − 0.229·19-s + 1.25·23-s − 1/5·25-s + 0.962·27-s + 0.742·29-s + 1.43·31-s − 0.696·33-s + 0.328·37-s − 0.320·39-s + 0.304·43-s − 0.596·45-s + 1.02·47-s − 49-s + 0.840·51-s + 1.23·53-s + 1.07·55-s + 0.132·57-s − 0.650·59-s − 0.256·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5077 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.91861293968938, −12.27418823400884, −11.85808097142983, −11.48037779170056, −11.05449386133079, −10.61564613042460, −10.24649006135186, −9.529164796649832, −9.160505520135695, −8.851091011067415, −8.364293831823100, −7.874119506859458, −6.866177037726508, −6.740588122004570, −6.367222157082352, −5.784869485545167, −5.530843834809801, −4.671033113350510, −4.478106627342627, −3.834044649844308, −3.068516481374282, −2.601651639793313, −2.073326248936649, −1.255568783034300, −0.9152530544569560, 0,
0.9152530544569560, 1.255568783034300, 2.073326248936649, 2.601651639793313, 3.068516481374282, 3.834044649844308, 4.478106627342627, 4.671033113350510, 5.530843834809801, 5.784869485545167, 6.367222157082352, 6.740588122004570, 6.866177037726508, 7.874119506859458, 8.364293831823100, 8.851091011067415, 9.160505520135695, 9.529164796649832, 10.24649006135186, 10.61564613042460, 11.05449386133079, 11.48037779170056, 11.85808097142983, 12.27418823400884, 12.91861293968938
