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
−7.37953027768837106783962470187, −6.94902811346530516535084185164, −6.00349519945841203064216099872, −5.21505028067511115441520100545, −4.66377695270198579742645853657, −3.95932542186001261708726753823, −3.24081105297797690749435891521, −2.41580478207842296073782829612, −1.37854838140684239127386514280, 0,
1.37854838140684239127386514280, 2.41580478207842296073782829612, 3.24081105297797690749435891521, 3.95932542186001261708726753823, 4.66377695270198579742645853657, 5.21505028067511115441520100545, 6.00349519945841203064216099872, 6.94902811346530516535084185164, 7.37953027768837106783962470187