| L(s) = 1 | − 3·3-s + 2.58·5-s + 33.7·7-s + 9·9-s − 33.1·11-s + 4.33·13-s − 7.74·15-s + 11.1·17-s + 121.·19-s − 101.·21-s + 14.8·23-s − 118.·25-s − 27·27-s − 272.·29-s + 165.·31-s + 99.4·33-s + 87.1·35-s + 30.5·37-s − 12.9·39-s + 400.·41-s + 274.·43-s + 23.2·45-s + 487.·47-s + 795.·49-s − 33.4·51-s + 208.·53-s − 85.6·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.231·5-s + 1.82·7-s + 0.333·9-s − 0.909·11-s + 0.0924·13-s − 0.133·15-s + 0.159·17-s + 1.47·19-s − 1.05·21-s + 0.134·23-s − 0.946·25-s − 0.192·27-s − 1.74·29-s + 0.956·31-s + 0.524·33-s + 0.421·35-s + 0.135·37-s − 0.0533·39-s + 1.52·41-s + 0.974·43-s + 0.0770·45-s + 1.51·47-s + 2.32·49-s − 0.0919·51-s + 0.540·53-s − 0.210·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.982364661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.982364661\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| good | 5 | \( 1 - 2.58T + 125T^{2} \) |
| 7 | \( 1 - 33.7T + 343T^{2} \) |
| 11 | \( 1 + 33.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 4.33T + 2.19e3T^{2} \) |
| 17 | \( 1 - 11.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 121.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 14.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 272.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 165.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 30.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 400.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 274.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 487.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 208.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 369.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 411.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 407.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 262.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 562.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 955.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 669.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.32e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.10e3T + 9.12e5T^{2} \) |
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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
−11.11331561542235101812670898546, −10.17592501625511623832369960790, −9.110417312462082707171739149045, −7.82753522680243510083773974111, −7.47344331461720482634089467430, −5.73493302331704809753459702680, −5.24479892738574186395715362752, −4.12698650421892090156327409625, −2.30570369214990967730934655313, −1.01858093037683575308349803196,
1.01858093037683575308349803196, 2.30570369214990967730934655313, 4.12698650421892090156327409625, 5.24479892738574186395715362752, 5.73493302331704809753459702680, 7.47344331461720482634089467430, 7.82753522680243510083773974111, 9.110417312462082707171739149045, 10.17592501625511623832369960790, 11.11331561542235101812670898546
