| L(s) = 1 | + 3·3-s − 7-s + 6·9-s + 11-s − 7·13-s − 3·17-s + 8·19-s − 3·21-s + 23-s + 9·27-s − 5·29-s + 2·31-s + 3·33-s + 4·37-s − 21·39-s − 8·41-s + 6·43-s + 3·47-s + 49-s − 9·51-s − 2·53-s + 24·57-s − 2·59-s − 14·61-s − 6·63-s − 4·67-s + 3·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.377·7-s + 2·9-s + 0.301·11-s − 1.94·13-s − 0.727·17-s + 1.83·19-s − 0.654·21-s + 0.208·23-s + 1.73·27-s − 0.928·29-s + 0.359·31-s + 0.522·33-s + 0.657·37-s − 3.36·39-s − 1.24·41-s + 0.914·43-s + 0.437·47-s + 1/7·49-s − 1.26·51-s − 0.274·53-s + 3.17·57-s − 0.260·59-s − 1.79·61-s − 0.755·63-s − 0.488·67-s + 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.37952557366795, −14.06471968380906, −13.58117553975764, −13.13513911183850, −12.60260640100281, −11.99897858504935, −11.69987556986943, −10.75793458058346, −10.23605571311841, −9.541923689803721, −9.401535290697227, −9.089704832177703, −8.262901088232129, −7.707797259195066, −7.363427865776923, −6.999976874357989, −6.226963050976408, −5.370354410984448, −4.781940519757821, −4.245627210947264, −3.521021310863870, −2.995892049311852, −2.555593279530405, −1.940648161866249, −1.155475847430924, 0,
1.155475847430924, 1.940648161866249, 2.555593279530405, 2.995892049311852, 3.521021310863870, 4.245627210947264, 4.781940519757821, 5.370354410984448, 6.226963050976408, 6.999976874357989, 7.363427865776923, 7.707797259195066, 8.262901088232129, 9.089704832177703, 9.401535290697227, 9.541923689803721, 10.23605571311841, 10.75793458058346, 11.69987556986943, 11.99897858504935, 12.60260640100281, 13.13513911183850, 13.58117553975764, 14.06471968380906, 14.37952557366795
