| L(s) = 1 | − 2-s − 4-s − 2·5-s + 3·8-s + 2·10-s + 11-s + 2·13-s − 16-s − 2·17-s + 2·20-s − 22-s − 4·23-s − 25-s − 2·26-s + 29-s + 8·31-s − 5·32-s + 2·34-s + 6·37-s − 6·40-s − 10·41-s + 8·43-s − 44-s + 4·46-s + 8·47-s + 50-s − 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.894·5-s + 1.06·8-s + 0.632·10-s + 0.301·11-s + 0.554·13-s − 1/4·16-s − 0.485·17-s + 0.447·20-s − 0.213·22-s − 0.834·23-s − 1/5·25-s − 0.392·26-s + 0.185·29-s + 1.43·31-s − 0.883·32-s + 0.342·34-s + 0.986·37-s − 0.948·40-s − 1.56·41-s + 1.21·43-s − 0.150·44-s + 0.589·46-s + 1.16·47-s + 0.141·50-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140679 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140679 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.57728258557002, −13.37835090959933, −12.62696410482236, −12.01157884974960, −11.80158033233486, −11.21952437455855, −10.66395672540812, −10.22803309059514, −9.821584864376138, −9.148137202261560, −8.810267932951510, −8.241837842405254, −7.992859257158702, −7.443612840899927, −6.879343836798267, −6.331867226958451, −5.699197796761858, −5.100966216974453, −4.352197264710560, −4.078323685406493, −3.695842190934766, −2.776223322531337, −2.151753277471134, −1.268606567756548, −0.7444902331239467, 0,
0.7444902331239467, 1.268606567756548, 2.151753277471134, 2.776223322531337, 3.695842190934766, 4.078323685406493, 4.352197264710560, 5.100966216974453, 5.699197796761858, 6.331867226958451, 6.879343836798267, 7.443612840899927, 7.992859257158702, 8.241837842405254, 8.810267932951510, 9.148137202261560, 9.821584864376138, 10.22803309059514, 10.66395672540812, 11.21952437455855, 11.80158033233486, 12.01157884974960, 12.62696410482236, 13.37835090959933, 13.57728258557002
