| L(s) = 1 | + 3-s − 1.28i·5-s + 1.96i·7-s + 9-s − 5.51i·11-s − 1.28i·15-s − 1.82·17-s + 8.08i·19-s + 1.96i·21-s + 6.97·23-s + 3.34·25-s + 27-s − 2.22·29-s + 4.44i·31-s − 5.51i·33-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.575i·5-s + 0.740i·7-s + 0.333·9-s − 1.66i·11-s − 0.332i·15-s − 0.442·17-s + 1.85i·19-s + 0.427i·21-s + 1.45·23-s + 0.668·25-s + 0.192·27-s − 0.413·29-s + 0.798i·31-s − 0.960i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.460937246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.460937246\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 1.28iT - 5T^{2} \) |
| 7 | \( 1 - 1.96iT - 7T^{2} \) |
| 11 | \( 1 + 5.51iT - 11T^{2} \) |
| 17 | \( 1 + 1.82T + 17T^{2} \) |
| 19 | \( 1 - 8.08iT - 19T^{2} \) |
| 23 | \( 1 - 6.97T + 23T^{2} \) |
| 29 | \( 1 + 2.22T + 29T^{2} \) |
| 31 | \( 1 - 4.44iT - 31T^{2} \) |
| 37 | \( 1 - 2.80iT - 37T^{2} \) |
| 41 | \( 1 - 4.58iT - 41T^{2} \) |
| 43 | \( 1 - 5.39T + 43T^{2} \) |
| 47 | \( 1 + 5.05iT - 47T^{2} \) |
| 53 | \( 1 + 2.44T + 53T^{2} \) |
| 59 | \( 1 + 10.0iT - 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 + 2.39iT - 67T^{2} \) |
| 71 | \( 1 + 15.1iT - 71T^{2} \) |
| 73 | \( 1 + 2.62iT - 73T^{2} \) |
| 79 | \( 1 - 6.15T + 79T^{2} \) |
| 83 | \( 1 - 5.59iT - 83T^{2} \) |
| 89 | \( 1 + 15.0iT - 89T^{2} \) |
| 97 | \( 1 + 13.9iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.475472611465559353298170911660, −8.000027321137916089702340127536, −6.93962143603877509981705774362, −6.09443573064875019135203348121, −5.46320634639996086358627745815, −4.69718061441384927163994764787, −3.54331885276981336444897037057, −3.07971140870810128830133016271, −1.91918039504009686041051121014, −0.868042074206528444150684491004,
0.919116616979478201768076090094, 2.28968905733166634437059105252, 2.78410752033627454538115634491, 3.98584523059039425802942949881, 4.52351813026202965438366925123, 5.33613636579419910017740830501, 6.69850846697978529745207951716, 7.15209672267224405245645922166, 7.38003233355997670285491017695, 8.537714785728739693389724309972
