| L(s) = 1 | + 2-s − 4-s − 2·5-s − 5·7-s − 3·8-s − 2·10-s + 11-s − 2·13-s − 5·14-s − 16-s + 7·17-s + 2·20-s + 22-s − 23-s − 25-s − 2·26-s + 5·28-s + 3·29-s − 8·31-s + 5·32-s + 7·34-s + 10·35-s − 3·37-s + 6·40-s − 11·41-s − 9·43-s − 44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.88·7-s − 1.06·8-s − 0.632·10-s + 0.301·11-s − 0.554·13-s − 1.33·14-s − 1/4·16-s + 1.69·17-s + 0.447·20-s + 0.213·22-s − 0.208·23-s − 1/5·25-s − 0.392·26-s + 0.944·28-s + 0.557·29-s − 1.43·31-s + 0.883·32-s + 1.20·34-s + 1.69·35-s − 0.493·37-s + 0.948·40-s − 1.71·41-s − 1.37·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−11.76911925774876002323168977941, −10.09723345915266216293702502311, −9.589222861617679677472732936062, −8.456654271075751034104031309554, −7.24010762516308694595094620804, −6.20890544963687991112230285397, −5.13510871976709382233695472657, −3.67816481691677016709564523384, −3.28065995132021916187600985135, 0,
3.28065995132021916187600985135, 3.67816481691677016709564523384, 5.13510871976709382233695472657, 6.20890544963687991112230285397, 7.24010762516308694595094620804, 8.456654271075751034104031309554, 9.589222861617679677472732936062, 10.09723345915266216293702502311, 11.76911925774876002323168977941
