L(s) = 1 | + 3-s − 5-s + 9-s − 4·11-s + 2·13-s − 15-s + 2·17-s + 8·19-s + 4·23-s + 25-s + 27-s + 6·29-s − 4·33-s − 2·37-s + 2·39-s + 6·41-s − 4·43-s − 45-s + 12·47-s + 2·51-s + 6·53-s + 4·55-s + 8·57-s + 12·59-s + 14·61-s − 2·65-s + 12·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s + 1.83·19-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.696·33-s − 0.328·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s + 1.75·47-s + 0.280·51-s + 0.824·53-s + 0.539·55-s + 1.05·57-s + 1.56·59-s + 1.79·61-s − 0.248·65-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.399267210\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.399267210\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.46195760776831, −14.12997002794430, −13.59208788692566, −13.04325409832029, −12.71647699382326, −11.90089576287367, −11.63483538554291, −10.90804412047257, −10.39457798778961, −9.919879338493593, −9.376676186937139, −8.661462702467411, −8.292810611675570, −7.752370351058495, −7.170746254570789, −6.865055542737939, −5.797924108148493, −5.347243225159647, −4.876938440095991, −3.961389703061032, −3.527075319322247, −2.764974293722887, −2.446282971720374, −1.198118555513609, −0.7275193028648438,
0.7275193028648438, 1.198118555513609, 2.446282971720374, 2.764974293722887, 3.527075319322247, 3.961389703061032, 4.876938440095991, 5.347243225159647, 5.797924108148493, 6.865055542737939, 7.170746254570789, 7.752370351058495, 8.292810611675570, 8.661462702467411, 9.376676186937139, 9.919879338493593, 10.39457798778961, 10.90804412047257, 11.63483538554291, 11.90089576287367, 12.71647699382326, 13.04325409832029, 13.59208788692566, 14.12997002794430, 14.46195760776831