| L(s) = 1 | − 3-s − 2·4-s − 3·5-s − 2·9-s + 2·12-s + 3·15-s + 4·16-s + 6·20-s − 9·23-s + 4·25-s + 5·27-s − 5·31-s + 4·36-s + 7·37-s + 6·45-s − 12·47-s − 4·48-s − 7·49-s + 6·53-s − 15·59-s − 6·60-s − 8·64-s + 13·67-s + 9·69-s − 3·71-s − 4·75-s − 12·80-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 1.34·5-s − 2/3·9-s + 0.577·12-s + 0.774·15-s + 16-s + 1.34·20-s − 1.87·23-s + 4/5·25-s + 0.962·27-s − 0.898·31-s + 2/3·36-s + 1.15·37-s + 0.894·45-s − 1.75·47-s − 0.577·48-s − 49-s + 0.824·53-s − 1.95·59-s − 0.774·60-s − 64-s + 1.58·67-s + 1.08·69-s − 0.356·71-s − 0.461·75-s − 1.34·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−12.76419543201372417939560661413, −11.90069246142032537610481501027, −11.10044716470991944961809074243, −9.790262208323322424254691277923, −8.487357851440445247270968425966, −7.74971489374349908553225834156, −6.04214609726225978583223790134, −4.73070320716804110076258258958, −3.60577326155639270878357097593, 0,
3.60577326155639270878357097593, 4.73070320716804110076258258958, 6.04214609726225978583223790134, 7.74971489374349908553225834156, 8.487357851440445247270968425966, 9.790262208323322424254691277923, 11.10044716470991944961809074243, 11.90069246142032537610481501027, 12.76419543201372417939560661413
