| L(s) = 1 | − 2-s + 4-s + 2·5-s + 7-s − 8-s − 2·10-s − 2·13-s − 14-s + 16-s + 4·17-s + 4·19-s + 2·20-s − 5·23-s − 25-s + 2·26-s + 28-s + 29-s − 5·31-s − 32-s − 4·34-s + 2·35-s − 6·37-s − 4·38-s − 2·40-s − 9·41-s − 4·43-s + 5·46-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 − 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.917·19-s + 0.447·20-s − 1.04·23-s − 1/5·25-s + 0.392·26-s + 0.188·28-s + 0.185·29-s − 0.898·31-s − 0.176·32-s − 0.685·34-s + 0.338·35-s − 0.986·37-s − 0.648·38-s − 0.316·40-s − 1.40·41-s − 0.609·43-s + 0.737·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.62755261142168303949540022299, −6.88802458493184285277510474214, −6.14325451372748017568782338927, −5.43157422656459406455046647009, −4.95091116425360540527758330007, −3.70740136485391501105453337448, −2.96391308444643070628228521720, −1.91074101829439392317445256218, −1.44604833793318247353905997475, 0,
1.44604833793318247353905997475, 1.91074101829439392317445256218, 2.96391308444643070628228521720, 3.70740136485391501105453337448, 4.95091116425360540527758330007, 5.43157422656459406455046647009, 6.14325451372748017568782338927, 6.88802458493184285277510474214, 7.62755261142168303949540022299
