| L(s) = 1 | − 2-s + 3·3-s − 4-s − 3·6-s − 7-s + 3·8-s + 6·9-s − 3·12-s − 6·13-s + 14-s − 16-s − 6·18-s − 4·19-s − 3·21-s − 2·23-s + 9·24-s + 6·26-s + 9·27-s + 28-s − 29-s + 8·31-s − 5·32-s − 6·36-s + 3·37-s + 4·38-s − 18·39-s − 10·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s − 1/2·4-s − 1.22·6-s − 0.377·7-s + 1.06·8-s + 2·9-s − 0.866·12-s − 1.66·13-s + 0.267·14-s − 1/4·16-s − 1.41·18-s − 0.917·19-s − 0.654·21-s − 0.417·23-s + 1.83·24-s + 1.17·26-s + 1.73·27-s + 0.188·28-s − 0.185·29-s + 1.43·31-s − 0.883·32-s − 36-s + 0.493·37-s + 0.648·38-s − 2.88·39-s − 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 - p T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−15.74178957822650, −15.02249862702396, −14.94133344784081, −14.12039540900312, −13.82253240650867, −13.32431411883132, −12.57707691762935, −12.41031110038487, −11.39336943329755, −10.49438247840483, −9.903385738805053, −9.795689216511230, −9.145557890237841, −8.627665914858006, −8.025041317701337, −7.792963342268358, −7.006048428331787, −6.524848893795593, −5.288229320879117, −4.656614330108006, −4.050137665212316, −3.454757558165568, −2.410352387566078, −2.270132689098537, −1.115080751302585, 0,
1.115080751302585, 2.270132689098537, 2.410352387566078, 3.454757558165568, 4.050137665212316, 4.656614330108006, 5.288229320879117, 6.524848893795593, 7.006048428331787, 7.792963342268358, 8.025041317701337, 8.627665914858006, 9.145557890237841, 9.795689216511230, 9.903385738805053, 10.49438247840483, 11.39336943329755, 12.41031110038487, 12.57707691762935, 13.32431411883132, 13.82253240650867, 14.12039540900312, 14.94133344784081, 15.02249862702396, 15.74178957822650
