L(s) = 1 | + (−6.14e4 + 2.29e4i)2-s − 4.43e7i·3-s + (3.24e9 − 2.81e9i)4-s + 1.39e11·5-s + (1.01e12 + 2.72e12i)6-s + 2.23e13i·7-s + (−1.34e14 + 2.47e14i)8-s − 1.15e14·9-s + (−8.54e15 + 3.18e15i)10-s + 1.78e16i·11-s + (−1.24e17 − 1.43e17i)12-s + 2.16e17·13-s + (−5.11e17 − 1.37e18i)14-s − 6.17e18i·15-s + (2.61e18 − 1.82e19i)16-s − 6.14e19·17-s + ⋯ |
L(s) = 1 | + (−0.936 + 0.349i)2-s − 1.03i·3-s + (0.755 − 0.655i)4-s + 0.912·5-s + (0.360 + 0.965i)6-s + 0.672i·7-s + (−0.478 + 0.877i)8-s − 0.0623·9-s + (−0.854 + 0.318i)10-s + 0.389i·11-s + (−0.675 − 0.778i)12-s + 0.325·13-s + (−0.235 − 0.629i)14-s − 0.940i·15-s + (0.141 − 0.989i)16-s − 1.26·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{33}{2})\) |
\(\approx\) |
\(1.375112421\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.375112421\) |
\(L(17)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (6.14e4 - 2.29e4i)T \) |
good | 3 | \( 1 + 4.43e7iT - 1.85e15T^{2} \) |
| 5 | \( 1 - 1.39e11T + 2.32e22T^{2} \) |
| 7 | \( 1 - 2.23e13iT - 1.10e27T^{2} \) |
| 11 | \( 1 - 1.78e16iT - 2.11e33T^{2} \) |
| 13 | \( 1 - 2.16e17T + 4.42e35T^{2} \) |
| 17 | \( 1 + 6.14e19T + 2.36e39T^{2} \) |
| 19 | \( 1 - 4.44e20iT - 8.31e40T^{2} \) |
| 23 | \( 1 - 8.74e21iT - 3.76e43T^{2} \) |
| 29 | \( 1 - 4.53e23T + 6.26e46T^{2} \) |
| 31 | \( 1 - 1.07e24iT - 5.29e47T^{2} \) |
| 37 | \( 1 + 9.58e23T + 1.52e50T^{2} \) |
| 41 | \( 1 + 1.42e25T + 4.06e51T^{2} \) |
| 43 | \( 1 + 1.36e26iT - 1.86e52T^{2} \) |
| 47 | \( 1 - 1.63e26iT - 3.21e53T^{2} \) |
| 53 | \( 1 - 6.59e27T + 1.50e55T^{2} \) |
| 59 | \( 1 + 2.05e26iT - 4.64e56T^{2} \) |
| 61 | \( 1 - 1.66e28T + 1.35e57T^{2} \) |
| 67 | \( 1 - 2.31e29iT - 2.71e58T^{2} \) |
| 71 | \( 1 + 3.68e29iT - 1.73e59T^{2} \) |
| 73 | \( 1 - 2.79e29T + 4.22e59T^{2} \) |
| 79 | \( 1 - 2.19e30iT - 5.29e60T^{2} \) |
| 83 | \( 1 + 3.91e30iT - 2.57e61T^{2} \) |
| 89 | \( 1 + 2.76e31T + 2.40e62T^{2} \) |
| 97 | \( 1 - 2.89e30T + 3.77e63T^{2} \) |
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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
−17.57191090773405555766521575922, −15.66712305886866827198327348388, −13.81451990529279830590962740523, −12.10197142163130178914594355823, −10.09163319222092994240070628247, −8.552326881848522267617333126687, −6.94073657568444553624400773392, −5.76051166789296085697664462658, −2.19917284530657578357194319583, −1.32574118559947280449017266610,
0.65727447862748033665491306779, 2.49280493029866675264390903314, 4.32340190530278279496337715737, 6.62721960008975960967056576788, 8.807286009869433778354670440536, 10.02937510399076036663637165710, 11.02434028292786092435359036495, 13.36973821403625393190237935029, 15.54197665567965471081505911932, 16.81608746611385388807454326968