| L(s) = 1 | − 2-s − 3·3-s − 7·4-s + 3·6-s + 4·7-s + 15·8-s + 9·9-s − 68·11-s + 21·12-s − 82·13-s − 4·14-s + 41·16-s − 86·17-s − 9·18-s + 19·19-s − 12·21-s + 68·22-s + 18·23-s − 45·24-s + 82·26-s − 27·27-s − 28·28-s + 30·29-s − 298·31-s − 161·32-s + 204·33-s + 86·34-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 0.577·3-s − 7/8·4-s + 0.204·6-s + 0.215·7-s + 0.662·8-s + 1/3·9-s − 1.86·11-s + 0.505·12-s − 1.74·13-s − 0.0763·14-s + 0.640·16-s − 1.22·17-s − 0.117·18-s + 0.229·19-s − 0.124·21-s + 0.658·22-s + 0.163·23-s − 0.382·24-s + 0.618·26-s − 0.192·27-s − 0.188·28-s + 0.192·29-s − 1.72·31-s − 0.889·32-s + 1.07·33-s + 0.433·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - p T \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 68 T + p^{3} T^{2} \) |
| 13 | \( 1 + 82 T + p^{3} T^{2} \) |
| 17 | \( 1 + 86 T + p^{3} T^{2} \) |
| 23 | \( 1 - 18 T + p^{3} T^{2} \) |
| 29 | \( 1 - 30 T + p^{3} T^{2} \) |
| 31 | \( 1 + 298 T + p^{3} T^{2} \) |
| 37 | \( 1 - 34 T + p^{3} T^{2} \) |
| 41 | \( 1 - 52 T + p^{3} T^{2} \) |
| 43 | \( 1 + 482 T + p^{3} T^{2} \) |
| 47 | \( 1 - 114 T + p^{3} T^{2} \) |
| 53 | \( 1 + 362 T + p^{3} T^{2} \) |
| 59 | \( 1 + 210 T + p^{3} T^{2} \) |
| 61 | \( 1 + 718 T + p^{3} T^{2} \) |
| 67 | \( 1 - 904 T + p^{3} T^{2} \) |
| 71 | \( 1 + 988 T + p^{3} T^{2} \) |
| 73 | \( 1 - 488 T + p^{3} T^{2} \) |
| 79 | \( 1 + 530 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1032 T + p^{3} T^{2} \) |
| 89 | \( 1 + 880 T + p^{3} T^{2} \) |
| 97 | \( 1 + 246 T + p^{3} T^{2} \) |
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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
−8.303904225435382578141588312731, −7.63658211570533949103562651216, −6.95737333029022808985019818402, −5.55685281590411326916815134504, −4.99920887032309436520352025308, −4.43597035256288652302310283057, −2.95275784224818347832220648653, −1.82295564005590077081690964400, 0, 0,
1.82295564005590077081690964400, 2.95275784224818347832220648653, 4.43597035256288652302310283057, 4.99920887032309436520352025308, 5.55685281590411326916815134504, 6.95737333029022808985019818402, 7.63658211570533949103562651216, 8.303904225435382578141588312731
