| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 4·11-s + 12-s − 15-s + 16-s + 2·17-s − 18-s − 20-s − 4·22-s − 4·23-s − 24-s + 25-s + 27-s + 29-s + 30-s + 8·31-s − 32-s + 4·33-s − 2·34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.223·20-s − 0.852·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.185·29-s + 0.182·30-s + 1.43·31-s − 0.176·32-s + 0.696·33-s − 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.81468341855920, −13.08308149042978, −12.40960486940717, −12.25153863279134, −11.53054760896434, −11.32544322866877, −10.71323213271829, −9.955480490245809, −9.765035770295598, −9.312633320354084, −8.740266434963216, −8.177242968640562, −7.916970961594308, −7.484619872514439, −6.661506511141920, −6.388504465245359, −5.977421421944650, −4.939102400459161, −4.536417869386891, −3.914967059028595, −3.296588806374734, −2.915478743733923, −2.087199083092955, −1.460766382834137, −0.9395074020816947, 0,
0.9395074020816947, 1.460766382834137, 2.087199083092955, 2.915478743733923, 3.296588806374734, 3.914967059028595, 4.536417869386891, 4.939102400459161, 5.977421421944650, 6.388504465245359, 6.661506511141920, 7.484619872514439, 7.916970961594308, 8.177242968640562, 8.740266434963216, 9.312633320354084, 9.765035770295598, 9.955480490245809, 10.71323213271829, 11.32544322866877, 11.53054760896434, 12.25153863279134, 12.40960486940717, 13.08308149042978, 13.81468341855920
