| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 4·7-s − 8-s + 9-s − 11-s − 12-s + 4·14-s + 16-s − 2·17-s − 18-s + 19-s + 4·21-s + 22-s − 2·23-s + 24-s − 5·25-s − 27-s − 4·28-s + 10·29-s − 2·31-s − 32-s + 33-s + 2·34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 1.06·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.229·19-s + 0.872·21-s + 0.213·22-s − 0.417·23-s + 0.204·24-s − 25-s − 0.192·27-s − 0.755·28-s + 1.85·29-s − 0.359·31-s − 0.176·32-s + 0.174·33-s + 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6004091680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6004091680\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.592472309668874924894894158066, −9.193634537030711191031976589522, −8.015105138192102002303708567947, −7.23133595769312761876945392762, −6.30732794327755624588542937743, −5.92107479072607268909853743268, −4.55784588257165667053773261181, −3.39853786914680186073742917184, −2.33727792920545830526310526613, −0.62791044201569185474029367226,
0.62791044201569185474029367226, 2.33727792920545830526310526613, 3.39853786914680186073742917184, 4.55784588257165667053773261181, 5.92107479072607268909853743268, 6.30732794327755624588542937743, 7.23133595769312761876945392762, 8.015105138192102002303708567947, 9.193634537030711191031976589522, 9.592472309668874924894894158066
