| L(s) = 1 | − 0.629·3-s − 5·5-s + 7·7-s − 26.6·9-s − 18.3·11-s + 57.3·13-s + 3.14·15-s − 28.2·17-s − 86.2·19-s − 4.40·21-s + 87.2·23-s + 25·25-s + 33.7·27-s − 97.3·29-s − 144.·31-s + 11.5·33-s − 35·35-s + 322.·37-s − 36.1·39-s − 152.·41-s − 341.·43-s + 133.·45-s + 471.·47-s + 49·49-s + 17.8·51-s − 144.·53-s + 91.7·55-s + ⋯ |
| L(s) = 1 | − 0.121·3-s − 0.447·5-s + 0.377·7-s − 0.985·9-s − 0.502·11-s + 1.22·13-s + 0.0542·15-s − 0.403·17-s − 1.04·19-s − 0.0458·21-s + 0.790·23-s + 0.200·25-s + 0.240·27-s − 0.623·29-s − 0.837·31-s + 0.0609·33-s − 0.169·35-s + 1.43·37-s − 0.148·39-s − 0.581·41-s − 1.21·43-s + 0.440·45-s + 1.46·47-s + 0.142·49-s + 0.0488·51-s − 0.374·53-s + 0.224·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.407954931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.407954931\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 - 7T \) |
| good | 3 | \( 1 + 0.629T + 27T^{2} \) |
| 11 | \( 1 + 18.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 57.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 28.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 86.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 87.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 97.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 144.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 322.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 152.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 341.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 471.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 144.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 740.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 681.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 983.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 271.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 363.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 135.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.37e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 978.T + 9.12e5T^{2} \) |
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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
−9.248122400497556690577326945674, −8.540782147634410143698575178456, −8.012528085481208402100395123035, −6.92736385050583236275078335066, −6.03089039524881865243839603868, −5.22719486275794072442957874961, −4.18243955903396944287750225838, −3.22618757011526453473474669368, −2.07119750145928598415624579099, −0.60742284324802278546523222153,
0.60742284324802278546523222153, 2.07119750145928598415624579099, 3.22618757011526453473474669368, 4.18243955903396944287750225838, 5.22719486275794072442957874961, 6.03089039524881865243839603868, 6.92736385050583236275078335066, 8.012528085481208402100395123035, 8.540782147634410143698575178456, 9.248122400497556690577326945674
