| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s − 11-s + 2·13-s − 15-s + 2·17-s − 4·19-s + 21-s + 25-s + 27-s + 2·29-s − 8·31-s − 33-s − 35-s + 10·37-s + 2·39-s − 6·41-s + 4·43-s − 45-s + 49-s + 2·51-s + 2·53-s + 55-s − 4·57-s + 12·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s + 0.218·21-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.174·33-s − 0.169·35-s + 1.64·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s + 1/7·49-s + 0.280·51-s + 0.274·53-s + 0.134·55-s − 0.529·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−14.51679369896404, −13.81479775517164, −13.34485182479615, −12.83388437857657, −12.48092522116902, −11.80640777527033, −11.24313102787237, −10.93803556630887, −10.18128193501785, −9.957385573948437, −8.988854800099662, −8.793849683763373, −8.267428765145051, −7.592651053899180, −7.403546484183258, −6.645091789152236, −5.939356600349499, −5.562663093515332, −4.576030373323387, −4.384274272956419, −3.603333977371895, −3.107434916317376, −2.392961560292524, −1.722491922915457, −0.9976943221461198, 0,
0.9976943221461198, 1.722491922915457, 2.392961560292524, 3.107434916317376, 3.603333977371895, 4.384274272956419, 4.576030373323387, 5.562663093515332, 5.939356600349499, 6.645091789152236, 7.403546484183258, 7.592651053899180, 8.267428765145051, 8.793849683763373, 8.988854800099662, 9.957385573948437, 10.18128193501785, 10.93803556630887, 11.24313102787237, 11.80640777527033, 12.48092522116902, 12.83388437857657, 13.34485182479615, 13.81479775517164, 14.51679369896404
