L(s) = 1 | + (−1.65 + 0.523i)3-s + (−0.841 + 1.45i)5-s + (1.65 − 2.06i)7-s + (2.45 − 1.72i)9-s + (−0.622 − 1.07i)11-s + (1.96 + 3.39i)13-s + (0.626 − 2.84i)15-s + (−1.62 + 2.81i)17-s + (2.36 + 4.09i)19-s + (−1.65 + 4.27i)21-s + (0.199 − 0.344i)23-s + (1.08 + 1.87i)25-s + (−3.14 + 4.13i)27-s + (−3.19 + 5.54i)29-s − 0.578·31-s + ⋯ |
L(s) = 1 | + (−0.953 + 0.302i)3-s + (−0.376 + 0.651i)5-s + (0.625 − 0.780i)7-s + (0.817 − 0.576i)9-s + (−0.187 − 0.325i)11-s + (0.543 + 0.941i)13-s + (0.161 − 0.735i)15-s + (−0.394 + 0.683i)17-s + (0.541 + 0.938i)19-s + (−0.360 + 0.932i)21-s + (0.0415 − 0.0718i)23-s + (0.216 + 0.375i)25-s + (−0.604 + 0.796i)27-s + (−0.594 + 1.02i)29-s − 0.103·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.294 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.779688 + 0.575338i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.779688 + 0.575338i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.65 - 0.523i)T \) |
| 7 | \( 1 + (-1.65 + 2.06i)T \) |
good | 5 | \( 1 + (0.841 - 1.45i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.622 + 1.07i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.96 - 3.39i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.62 - 2.81i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.36 - 4.09i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.199 + 0.344i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.19 - 5.54i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 0.578T + 31T^{2} \) |
| 37 | \( 1 + (-2.72 - 4.71i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.20 - 7.27i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.46 + 4.26i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 0.425T + 47T^{2} \) |
| 53 | \( 1 + (0.466 - 0.807i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 6.05T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 + 9.41T + 67T^{2} \) |
| 71 | \( 1 - 8.46T + 71T^{2} \) |
| 73 | \( 1 + (-6.82 + 11.8i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 5.53T + 79T^{2} \) |
| 83 | \( 1 + (8.03 - 13.9i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (6.03 + 10.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.86 - 10.1i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.09370575229707788936836405950, −10.53418401183851071231571971299, −9.546298015547297928024611303920, −8.315438059145772213720786857482, −7.27792345417610486490677924642, −6.55436693430683961950249079657, −5.49826233590120266389478958190, −4.33497383575488485478498802825, −3.54753115521460238712110689257, −1.41651964553109203068790263629,
0.73841585416133661190646578768, 2.40625780442605696993435953406, 4.28942025281433834325958726840, 5.18987799422661449008420135559, 5.84459020586953351149653413480, 7.14808737396611332281141241671, 7.971567434603356229456123450386, 8.887932014539684168905828554504, 9.909017216838674930621341008450, 11.13964847501536052845923594188