| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 11-s + 4·13-s − 15-s + 17-s + 4·19-s + 21-s + 9·23-s + 25-s − 27-s − 9·29-s − 31-s − 33-s − 35-s − 2·37-s − 4·39-s + 12·41-s + 43-s + 45-s − 6·49-s − 51-s + 55-s − 4·57-s − 6·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.258·15-s + 0.242·17-s + 0.917·19-s + 0.218·21-s + 1.87·23-s + 1/5·25-s − 0.192·27-s − 1.67·29-s − 0.179·31-s − 0.174·33-s − 0.169·35-s − 0.328·37-s − 0.640·39-s + 1.87·41-s + 0.152·43-s + 0.149·45-s − 6/7·49-s − 0.140·51-s + 0.134·55-s − 0.529·57-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 179520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.334488431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.334488431\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−13.18687785756739, −12.78304367194164, −12.27699038726064, −11.64891952780968, −11.17551589274926, −10.94611687969151, −10.44006080317806, −9.790766378607749, −9.247702954344682, −9.131189028668289, −8.519052413726645, −7.548241547521650, −7.520243826666673, −6.759907857692354, −6.323540281472035, −5.771716548285576, −5.470241706175978, −4.879403139923166, −4.216082957568063, −3.644902461535979, −3.111910172589947, −2.581203831973027, −1.478670732089192, −1.340854128162238, −0.4806371768909870,
0.4806371768909870, 1.340854128162238, 1.478670732089192, 2.581203831973027, 3.111910172589947, 3.644902461535979, 4.216082957568063, 4.879403139923166, 5.470241706175978, 5.771716548285576, 6.323540281472035, 6.759907857692354, 7.520243826666673, 7.548241547521650, 8.519052413726645, 9.131189028668289, 9.247702954344682, 9.790766378607749, 10.44006080317806, 10.94611687969151, 11.17551589274926, 11.64891952780968, 12.27699038726064, 12.78304367194164, 13.18687785756739
