| L(s) = 1 | + 2-s + 4-s + 2·5-s + 7-s + 8-s + 2·10-s + 4·13-s + 14-s + 16-s + 2·20-s − 6·23-s − 25-s + 4·26-s + 28-s + 4·29-s + 32-s + 2·35-s + 2·37-s + 2·40-s + 10·41-s − 4·43-s − 6·46-s + 12·47-s + 49-s − 50-s + 4·52-s − 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.377·7-s + 0.353·8-s + 0.632·10-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.447·20-s − 1.25·23-s − 1/5·25-s + 0.784·26-s + 0.188·28-s + 0.742·29-s + 0.176·32-s + 0.338·35-s + 0.328·37-s + 0.316·40-s + 1.56·41-s − 0.609·43-s − 0.884·46-s + 1.75·47-s + 1/7·49-s − 0.141·50-s + 0.554·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.707818308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.707818308\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−14.74767941640725, −14.16983881825876, −13.89503675009333, −13.45309900432351, −12.95033419176974, −12.25020559901975, −11.91919246146014, −11.17561350822240, −10.73321078280288, −10.27852057887324, −9.560421728419597, −9.139572583292205, −8.303612106941297, −7.948049240882987, −7.211114822040266, −6.449882394106633, −6.011255204860187, −5.667668235815969, −4.931535003381408, −4.204753911401759, −3.785508264069041, −2.889636411224629, −2.226598968104275, −1.627665643691601, −0.8039573952475534,
0.8039573952475534, 1.627665643691601, 2.226598968104275, 2.889636411224629, 3.785508264069041, 4.204753911401759, 4.931535003381408, 5.667668235815969, 6.011255204860187, 6.449882394106633, 7.211114822040266, 7.948049240882987, 8.303612106941297, 9.139572583292205, 9.560421728419597, 10.27852057887324, 10.73321078280288, 11.17561350822240, 11.91919246146014, 12.25020559901975, 12.95033419176974, 13.45309900432351, 13.89503675009333, 14.16983881825876, 14.74767941640725
