| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 4·7-s + 8-s + 9-s + 11-s − 12-s − 13-s + 4·14-s + 16-s − 17-s + 18-s − 4·21-s + 22-s − 23-s − 24-s − 26-s − 27-s + 4·28-s − 2·29-s − 2·31-s + 32-s − 33-s − 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 − 0.277·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.872·21-s + 0.213·22-s − 0.208·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s + 0.755·28-s − 0.371·29-s − 0.359·31-s + 0.176·32-s − 0.174·33-s − 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 17 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.67607444748935, −12.18015463028406, −11.82576112036118, −11.46895789975823, −11.04758477106179, −10.68143846337456, −10.21837967863787, −9.661208332480877, −9.028714206616379, −8.632618832988432, −8.027898392070406, −7.590365591620446, −7.221182867453664, −6.691992609895456, −6.104666524588539, −5.650745089879349, −5.245225427847634, −4.699832495799361, −4.422955727223631, −3.866749444216608, −3.303333821175396, −2.497960073260472, −2.016845741070791, −1.497073353909728, −0.9295724219142868, 0,
0.9295724219142868, 1.497073353909728, 2.016845741070791, 2.497960073260472, 3.303333821175396, 3.866749444216608, 4.422955727223631, 4.699832495799361, 5.245225427847634, 5.650745089879349, 6.104666524588539, 6.691992609895456, 7.221182867453664, 7.590365591620446, 8.027898392070406, 8.632618832988432, 9.028714206616379, 9.661208332480877, 10.21837967863787, 10.68143846337456, 11.04758477106179, 11.46895789975823, 11.82576112036118, 12.18015463028406, 12.67607444748935
