| L(s) = 1 | + 5·5-s − 10.2·7-s + 44.3·11-s − 50.8·13-s + 25.2·17-s + 31.3·19-s − 76.1·23-s + 25·25-s + 156.·29-s − 134.·31-s − 51.2·35-s + 81.2·37-s + 326.·41-s − 422.·43-s + 452.·47-s − 237.·49-s + 98.1·53-s + 221.·55-s − 540.·59-s + 522.·61-s − 254.·65-s + 129.·67-s − 26.6·71-s − 147.·73-s − 454.·77-s + 1.08e3·79-s − 594.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.553·7-s + 1.21·11-s − 1.08·13-s + 0.360·17-s + 0.378·19-s − 0.690·23-s + 0.200·25-s + 0.998·29-s − 0.779·31-s − 0.247·35-s + 0.360·37-s + 1.24·41-s − 1.49·43-s + 1.40·47-s − 0.693·49-s + 0.254·53-s + 0.543·55-s − 1.19·59-s + 1.09·61-s − 0.485·65-s + 0.235·67-s − 0.0444·71-s − 0.237·73-s − 0.673·77-s + 1.55·79-s − 0.786·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.218144326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.218144326\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 + 10.2T + 343T^{2} \) |
| 11 | \( 1 - 44.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 50.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 25.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 31.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 76.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 156.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 134.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 81.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 326.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 422.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 452.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 98.1T + 1.48e5T^{2} \) |
| 59 | \( 1 + 540.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 522.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 129.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 26.6T + 3.57e5T^{2} \) |
| 73 | \( 1 + 147.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.08e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 594.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 592.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 666.T + 9.12e5T^{2} \) |
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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
−8.893264620884909463560142931877, −7.900709345842155104201924872688, −7.06946208982433766719888245251, −6.40353320112490616042641398705, −5.64643679931311375098650570035, −4.69883424859375951691219361431, −3.78712464588099908583269822776, −2.84300705714419785474799367710, −1.82454083217973335123660263679, −0.67727262051236517908123904869,
0.67727262051236517908123904869, 1.82454083217973335123660263679, 2.84300705714419785474799367710, 3.78712464588099908583269822776, 4.69883424859375951691219361431, 5.64643679931311375098650570035, 6.40353320112490616042641398705, 7.06946208982433766719888245251, 7.900709345842155104201924872688, 8.893264620884909463560142931877
