| L(s) = 1 | + 5-s + 7-s − 3·9-s − 2·11-s + 2·13-s + 3·17-s + 2·19-s + 23-s + 25-s − 7·29-s − 5·31-s + 35-s − 11·37-s + 41-s − 3·45-s − 6·49-s − 11·53-s − 2·55-s + 13·59-s + 8·61-s − 3·63-s + 2·65-s − 5·67-s + 5·71-s + 6·73-s − 2·77-s − 12·79-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 9-s − 0.603·11-s + 0.554·13-s + 0.727·17-s + 0.458·19-s + 0.208·23-s + 1/5·25-s − 1.29·29-s − 0.898·31-s + 0.169·35-s − 1.80·37-s + 0.156·41-s − 0.447·45-s − 6/7·49-s − 1.51·53-s − 0.269·55-s + 1.69·59-s + 1.02·61-s − 0.377·63-s + 0.248·65-s − 0.610·67-s + 0.593·71-s + 0.702·73-s − 0.227·77-s − 1.35·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−7.59653629758399943262516221184, −6.91143568407048423453135395071, −5.97543117230083726419167760169, −5.42647673030720855877781152709, −5.02637245760001494301036384371, −3.75006783743032921485299913724, −3.18806675388433944305091624836, −2.23093209652595357140500085124, −1.36861505540149021722314561914, 0,
1.36861505540149021722314561914, 2.23093209652595357140500085124, 3.18806675388433944305091624836, 3.75006783743032921485299913724, 5.02637245760001494301036384371, 5.42647673030720855877781152709, 5.97543117230083726419167760169, 6.91143568407048423453135395071, 7.59653629758399943262516221184
