L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 11-s + 2·13-s − 14-s + 16-s − 6·17-s − 4·19-s + 20-s − 22-s + 25-s − 2·26-s + 28-s − 6·29-s + 8·31-s − 32-s + 6·34-s + 35-s − 10·37-s + 4·38-s − 40-s + 6·41-s + 8·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.223·20-s − 0.213·22-s + 1/5·25-s − 0.392·26-s + 0.188·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s + 1.02·34-s + 0.169·35-s − 1.64·37-s + 0.648·38-s − 0.158·40-s + 0.937·41-s + 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.68213282272626176489605257304, −6.91401765119263120268516397948, −6.29673898546517376844981476190, −5.74055396629023604212769282772, −4.65938529091490651289305298981, −4.06263322163169348630900887540, −2.91889112157078355155712107243, −2.07696074368181796662597651540, −1.35555109562721555053754259757, 0,
1.35555109562721555053754259757, 2.07696074368181796662597651540, 2.91889112157078355155712107243, 4.06263322163169348630900887540, 4.65938529091490651289305298981, 5.74055396629023604212769282772, 6.29673898546517376844981476190, 6.91401765119263120268516397948, 7.68213282272626176489605257304