L(s) = 1 | − 2-s + 3·3-s + 4-s + 4·5-s − 3·6-s − 8-s + 6·9-s − 4·10-s − 11-s + 3·12-s + 13-s + 12·15-s + 16-s − 2·17-s − 6·18-s − 6·19-s + 4·20-s + 22-s − 2·23-s − 3·24-s + 11·25-s − 26-s + 9·27-s + 29-s − 12·30-s − 4·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s + 1.78·5-s − 1.22·6-s − 0.353·8-s + 2·9-s − 1.26·10-s − 0.301·11-s + 0.866·12-s + 0.277·13-s + 3.09·15-s + 1/4·16-s − 0.485·17-s − 1.41·18-s − 1.37·19-s + 0.894·20-s + 0.213·22-s − 0.417·23-s − 0.612·24-s + 11/5·25-s − 0.196·26-s + 1.73·27-s + 0.185·29-s − 2.19·30-s − 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.795943301\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.795943301\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−9.679219573419020754195198177157, −8.995602629584817296380306732145, −8.574808871033960266742160369202, −7.64097690685396419289660444604, −6.65817966824607617360279018406, −5.89723575121056028210205499246, −4.51301965109001578483282794481, −3.15391324087188614918799611445, −2.23521168334471743395036156035, −1.69694527723194007972518751957,
1.69694527723194007972518751957, 2.23521168334471743395036156035, 3.15391324087188614918799611445, 4.51301965109001578483282794481, 5.89723575121056028210205499246, 6.65817966824607617360279018406, 7.64097690685396419289660444604, 8.574808871033960266742160369202, 8.995602629584817296380306732145, 9.679219573419020754195198177157