| L(s) = 1 | − 243·9-s + 76·11-s + 4.95e3·23-s − 3.12e3·25-s + 7.28e3·29-s − 8.88e3·37-s − 1.17e4·43-s + 2.45e4·53-s − 6.93e4·67-s + 2.22e3·71-s − 8.01e4·79-s + 5.90e4·81-s − 1.84e4·99-s + 6.49e4·107-s − 2.19e5·109-s + 1.23e5·113-s + ⋯ |
| L(s) = 1 | − 9-s + 0.189·11-s + 1.95·23-s − 25-s + 1.60·29-s − 1.06·37-s − 0.968·43-s + 1.20·53-s − 1.88·67-s + 0.0523·71-s − 1.44·79-s + 81-s − 0.189·99-s + 0.548·107-s − 1.77·109-s + 0.907·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p^{5} T^{2} \) |
| 5 | \( 1 + p^{5} T^{2} \) |
| 11 | \( 1 - 76 T + p^{5} T^{2} \) |
| 13 | \( 1 + p^{5} T^{2} \) |
| 17 | \( 1 + p^{5} T^{2} \) |
| 19 | \( 1 + p^{5} T^{2} \) |
| 23 | \( 1 - 4952 T + p^{5} T^{2} \) |
| 29 | \( 1 - 7282 T + p^{5} T^{2} \) |
| 31 | \( 1 + p^{5} T^{2} \) |
| 37 | \( 1 + 8886 T + p^{5} T^{2} \) |
| 41 | \( 1 + p^{5} T^{2} \) |
| 43 | \( 1 + 11748 T + p^{5} T^{2} \) |
| 47 | \( 1 + p^{5} T^{2} \) |
| 53 | \( 1 - 24550 T + p^{5} T^{2} \) |
| 59 | \( 1 + p^{5} T^{2} \) |
| 61 | \( 1 + p^{5} T^{2} \) |
| 67 | \( 1 + 69364 T + p^{5} T^{2} \) |
| 71 | \( 1 - 2224 T + p^{5} T^{2} \) |
| 73 | \( 1 + p^{5} T^{2} \) |
| 79 | \( 1 + 80168 T + p^{5} T^{2} \) |
| 83 | \( 1 + p^{5} T^{2} \) |
| 89 | \( 1 + p^{5} T^{2} \) |
| 97 | \( 1 + p^{5} T^{2} \) |
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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
−8.927239268082452478690194704291, −8.468054977008571319468593971231, −7.34847590775182169497905549943, −6.49274819975240320581294009898, −5.54186473877240889952814440263, −4.69389896881028523496819430429, −3.42805135504980879472105824248, −2.58970239925352720994059983834, −1.22480681209890432698537454433, 0,
1.22480681209890432698537454433, 2.58970239925352720994059983834, 3.42805135504980879472105824248, 4.69389896881028523496819430429, 5.54186473877240889952814440263, 6.49274819975240320581294009898, 7.34847590775182169497905549943, 8.468054977008571319468593971231, 8.927239268082452478690194704291
