| L(s) = 1 | + (−0.766 − 0.642i)2-s + (1.44 + 0.948i)3-s + (0.173 + 0.984i)4-s + (−3.88 − 1.41i)5-s + (−0.500 − 1.65i)6-s + (0.173 − 0.984i)7-s + (0.500 − 0.866i)8-s + (1.19 + 2.74i)9-s + (2.06 + 3.58i)10-s + (3.52 − 1.28i)11-s + (−0.682 + 1.59i)12-s + (5.37 − 4.50i)13-s + (−0.766 + 0.642i)14-s + (−4.29 − 5.74i)15-s + (−0.939 + 0.342i)16-s + (−1.81 − 3.13i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (0.836 + 0.547i)3-s + (0.0868 + 0.492i)4-s + (−1.73 − 0.633i)5-s + (−0.204 − 0.676i)6-s + (0.0656 − 0.372i)7-s + (0.176 − 0.306i)8-s + (0.399 + 0.916i)9-s + (0.654 + 1.13i)10-s + (1.06 − 0.387i)11-s + (−0.197 + 0.459i)12-s + (1.49 − 1.25i)13-s + (−0.204 + 0.171i)14-s + (−1.10 − 1.48i)15-s + (−0.234 + 0.0855i)16-s + (−0.439 − 0.760i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.921354 - 0.564447i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.921354 - 0.564447i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 3 | \( 1 + (-1.44 - 0.948i)T \) |
| 7 | \( 1 + (-0.173 + 0.984i)T \) |
| good | 5 | \( 1 + (3.88 + 1.41i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (-3.52 + 1.28i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-5.37 + 4.50i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.81 + 3.13i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.707 + 1.22i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.927 + 5.25i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-4.73 - 3.97i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.433 + 2.45i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (1.48 + 2.57i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.14 - 3.48i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (8.36 - 3.04i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.610 + 3.45i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 4.71T + 53T^{2} \) |
| 59 | \( 1 + (2.09 + 0.761i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.00444 + 0.0252i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (3.51 - 2.94i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.85 - 3.21i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.10 + 3.65i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.27 - 4.42i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.57 - 5.51i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-5.32 + 9.22i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.58 + 2.03i)T + (74.3 - 62.3i)T^{2} \) |
| 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
−11.12835113802062408188341542014, −10.40559962310996959592712888686, −9.065132860833460997072311025448, −8.484029063132289240627193176987, −7.975783470440513988327261773210, −6.84135139319625533262708080761, −4.80266032141224157932511248234, −3.84255567616370948854417162413, −3.20424864446501670459392704977, −0.908086560565708861276579544818,
1.60771021407011348799989347172, 3.49659515932480003469873783205, 4.18256659938771432115700623434, 6.43116537209681908628115629738, 6.86119620165854411473709880834, 7.936206700416614033309204696073, 8.544891349692382521241152097671, 9.269230693689923606741130732334, 10.64566689694063428886665117342, 11.83563332434412634063742944444
