| L(s) = 1 | − 8·4-s + 89·13-s + 64·16-s + 56·19-s − 125·25-s − 289·31-s − 433·37-s + 71·43-s − 712·52-s + 719·61-s − 512·64-s + 1.00e3·67-s + 1.19e3·73-s − 448·76-s + 503·79-s − 523·97-s + 1.00e3·100-s − 19·103-s + 2.21e3·109-s + ⋯ |
| L(s) = 1 | − 4-s + 1.89·13-s + 16-s + 0.676·19-s − 25-s − 1.67·31-s − 1.92·37-s + 0.251·43-s − 1.89·52-s + 1.50·61-s − 64-s + 1.83·67-s + 1.90·73-s − 0.676·76-s + 0.716·79-s − 0.547·97-s + 100-s − 0.0181·103-s + 1.94·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.602105260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.602105260\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + p^{3} T^{2} \) |
| 5 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 - 89 T + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 - 56 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 + 289 T + p^{3} T^{2} \) |
| 37 | \( 1 + 433 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 - 71 T + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 - 719 T + p^{3} T^{2} \) |
| 67 | \( 1 - 1007 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 - 1190 T + p^{3} T^{2} \) |
| 79 | \( 1 - 503 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 + 523 T + p^{3} T^{2} \) |
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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
−9.163002706359222015692792715617, −8.548609339522697578618203263630, −7.84571362442804508379736964792, −6.77090679114611187415720676639, −5.73241800004350870429436957795, −5.17350018539273291030212347386, −3.82674854515148048284692065925, −3.54914775531436142843934480700, −1.78034677135038308101569687456, −0.66359926916394254861701684044,
0.66359926916394254861701684044, 1.78034677135038308101569687456, 3.54914775531436142843934480700, 3.82674854515148048284692065925, 5.17350018539273291030212347386, 5.73241800004350870429436957795, 6.77090679114611187415720676639, 7.84571362442804508379736964792, 8.548609339522697578618203263630, 9.163002706359222015692792715617
