| L(s) = 1 | − 3·2-s + 7·3-s + 4-s − 21·6-s + 22·7-s + 21·8-s + 22·9-s − 64·11-s + 7·12-s − 73·13-s − 66·14-s − 71·16-s + 17·17-s − 66·18-s − 49·19-s + 154·21-s + 192·22-s − 110·23-s + 147·24-s + 219·26-s − 35·27-s + 22·28-s + 155·29-s − 197·31-s + 45·32-s − 448·33-s − 51·34-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 1.34·3-s + 1/8·4-s − 1.42·6-s + 1.18·7-s + 0.928·8-s + 0.814·9-s − 1.75·11-s + 0.168·12-s − 1.55·13-s − 1.25·14-s − 1.10·16-s + 0.242·17-s − 0.864·18-s − 0.591·19-s + 1.60·21-s + 1.86·22-s − 0.997·23-s + 1.25·24-s + 1.65·26-s − 0.249·27-s + 0.148·28-s + 0.992·29-s − 1.14·31-s + 0.248·32-s − 2.36·33-s − 0.257·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 - p T \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 7 | \( 1 - 22 T + p^{3} T^{2} \) |
| 11 | \( 1 + 64 T + p^{3} T^{2} \) |
| 13 | \( 1 + 73 T + p^{3} T^{2} \) |
| 19 | \( 1 + 49 T + p^{3} T^{2} \) |
| 23 | \( 1 + 110 T + p^{3} T^{2} \) |
| 29 | \( 1 - 155 T + p^{3} T^{2} \) |
| 31 | \( 1 + 197 T + p^{3} T^{2} \) |
| 37 | \( 1 - 372 T + p^{3} T^{2} \) |
| 41 | \( 1 + 262 T + p^{3} T^{2} \) |
| 43 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 47 | \( 1 - 13 T + p^{3} T^{2} \) |
| 53 | \( 1 - 653 T + p^{3} T^{2} \) |
| 59 | \( 1 + 333 T + p^{3} T^{2} \) |
| 61 | \( 1 + 355 T + p^{3} T^{2} \) |
| 67 | \( 1 + 814 T + p^{3} T^{2} \) |
| 71 | \( 1 - 47 T + p^{3} T^{2} \) |
| 73 | \( 1 - 437 T + p^{3} T^{2} \) |
| 79 | \( 1 + 384 T + p^{3} T^{2} \) |
| 83 | \( 1 - 736 T + p^{3} T^{2} \) |
| 89 | \( 1 - 511 T + p^{3} T^{2} \) |
| 97 | \( 1 + 537 T + p^{3} T^{2} \) |
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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
−10.10750097761590647616916062583, −9.301041782307052843828931551690, −8.228856060749355784707420372375, −7.998304267290896809843369148345, −7.31278536293733836697034998003, −5.24989406333725030441642811869, −4.38325619236845914047215161386, −2.66393276515255032970062219503, −1.87728159533733311120663681774, 0,
1.87728159533733311120663681774, 2.66393276515255032970062219503, 4.38325619236845914047215161386, 5.24989406333725030441642811869, 7.31278536293733836697034998003, 7.998304267290896809843369148345, 8.228856060749355784707420372375, 9.301041782307052843828931551690, 10.10750097761590647616916062583
