L(s) = 1 | + 0.380·2-s + 1.90·3-s − 0.855·4-s + 0.0766·5-s + 0.726·6-s − 1.33·7-s − 0.706·8-s + 2.63·9-s + 0.0291·10-s + 1.79·11-s − 1.63·12-s − 0.506·14-s + 0.146·15-s + 0.586·16-s + 1.00·18-s − 0.0654·20-s − 2.53·21-s + 0.682·22-s − 1.34·24-s − 0.994·25-s + 3.11·27-s + 1.13·28-s − 1.94·29-s + 0.0556·30-s + 0.929·32-s + 3.41·33-s − 0.101·35-s + ⋯ |
L(s) = 1 | + 0.380·2-s + 1.90·3-s − 0.855·4-s + 0.0766·5-s + 0.726·6-s − 1.33·7-s − 0.706·8-s + 2.63·9-s + 0.0291·10-s + 1.79·11-s − 1.63·12-s − 0.506·14-s + 0.146·15-s + 0.586·16-s + 1.00·18-s − 0.0654·20-s − 2.53·21-s + 0.682·22-s − 1.34·24-s − 0.994·25-s + 3.11·27-s + 1.13·28-s − 1.94·29-s + 0.0556·30-s + 0.929·32-s + 3.41·33-s − 0.101·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.822089312\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.822089312\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1151 | \( 1 - T \) |
good | 2 | \( 1 - 0.380T + T^{2} \) |
| 3 | \( 1 - 1.90T + T^{2} \) |
| 5 | \( 1 - 0.0766T + T^{2} \) |
| 7 | \( 1 + 1.33T + T^{2} \) |
| 11 | \( 1 - 1.79T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.94T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.54T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.08T + T^{2} \) |
| 47 | \( 1 - 1.97T + T^{2} \) |
| 53 | \( 1 - 0.955T + T^{2} \) |
| 59 | \( 1 + 1.99T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.71T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 0.229T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.564061622103492956819259851540, −9.139920246060047542083903163390, −8.758515540627315180044721805470, −7.59938202790995349419882214499, −6.81304551709284438009563972955, −5.82796075586274958603373734139, −4.26834767264519063111684318160, −3.70743779098888958826730707371, −3.17362214526586055163500483110, −1.74418135004610338397050114911,
1.74418135004610338397050114911, 3.17362214526586055163500483110, 3.70743779098888958826730707371, 4.26834767264519063111684318160, 5.82796075586274958603373734139, 6.81304551709284438009563972955, 7.59938202790995349419882214499, 8.758515540627315180044721805470, 9.139920246060047542083903163390, 9.564061622103492956819259851540