L(s) = 1 | + 0.817·2-s − 5.32·3-s − 7.33·4-s + 5·5-s − 4.35·6-s + 30.6·7-s − 12.5·8-s + 1.40·9-s + 4.08·10-s − 11·11-s + 39.0·12-s − 31.7·13-s + 25.0·14-s − 26.6·15-s + 48.4·16-s + 98.1·17-s + 1.14·18-s + 19·19-s − 36.6·20-s − 163.·21-s − 8.99·22-s − 89.2·23-s + 66.7·24-s + 25·25-s − 25.9·26-s + 136.·27-s − 224.·28-s + ⋯ |
L(s) = 1 | + 0.288·2-s − 1.02·3-s − 0.916·4-s + 0.447·5-s − 0.296·6-s + 1.65·7-s − 0.553·8-s + 0.0519·9-s + 0.129·10-s − 0.301·11-s + 0.940·12-s − 0.676·13-s + 0.477·14-s − 0.458·15-s + 0.756·16-s + 1.40·17-s + 0.0150·18-s + 0.229·19-s − 0.409·20-s − 1.69·21-s − 0.0871·22-s − 0.808·23-s + 0.567·24-s + 0.200·25-s − 0.195·26-s + 0.972·27-s − 1.51·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.400183304\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.400183304\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 - 19T \) |
good | 2 | \( 1 - 0.817T + 8T^{2} \) |
| 3 | \( 1 + 5.32T + 27T^{2} \) |
| 7 | \( 1 - 30.6T + 343T^{2} \) |
| 13 | \( 1 + 31.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 98.1T + 4.91e3T^{2} \) |
| 23 | \( 1 + 89.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 33.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 64.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 217.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 430.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 472.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 271.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 249.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 448.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 411.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 274.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 936.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 723.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 181.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 858.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 758.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 294.T + 9.12e5T^{2} \) |
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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
−9.863801789431306708013475503130, −8.450014647733936922031863463301, −8.129005156232646014919400479472, −6.90413414671901514894619720636, −5.63431784473177629071157795183, −5.23787860726851336476139230781, −4.72686312281770334445088044887, −3.42248709202638879089307419410, −1.84463809955383421780114974350, −0.64951090991392475280698687498,
0.64951090991392475280698687498, 1.84463809955383421780114974350, 3.42248709202638879089307419410, 4.72686312281770334445088044887, 5.23787860726851336476139230781, 5.63431784473177629071157795183, 6.90413414671901514894619720636, 8.129005156232646014919400479472, 8.450014647733936922031863463301, 9.863801789431306708013475503130