| L(s) = 1 | + (−1.57 + 4.85i)2-s + (6.28 + 4.56i)3-s + (−14.6 − 10.6i)4-s + (6.03 − 9.41i)5-s + (−32.0 + 23.2i)6-s − 5.45·7-s + (41.5 − 30.1i)8-s + (10.2 + 31.6i)9-s + (36.1 + 44.1i)10-s + (8.51 − 26.2i)11-s + (−43.3 − 133. i)12-s + (10.5 + 32.3i)13-s + (8.60 − 26.4i)14-s + (80.8 − 31.5i)15-s + (36.3 + 111. i)16-s + (−50.5 + 36.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.557 + 1.71i)2-s + (1.20 + 0.878i)3-s + (−1.82 − 1.32i)4-s + (0.539 − 0.841i)5-s + (−2.18 + 1.58i)6-s − 0.294·7-s + (1.83 − 1.33i)8-s + (0.380 + 1.17i)9-s + (1.14 + 1.39i)10-s + (0.233 − 0.718i)11-s + (−1.04 − 3.20i)12-s + (0.224 + 0.690i)13-s + (0.164 − 0.505i)14-s + (1.39 − 0.543i)15-s + (0.567 + 1.74i)16-s + (−0.721 + 0.524i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.630 - 0.776i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.499765 + 1.05000i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.499765 + 1.05000i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-6.03 + 9.41i)T \) |
| good | 2 | \( 1 + (1.57 - 4.85i)T + (-6.47 - 4.70i)T^{2} \) |
| 3 | \( 1 + (-6.28 - 4.56i)T + (8.34 + 25.6i)T^{2} \) |
| 7 | \( 1 + 5.45T + 343T^{2} \) |
| 11 | \( 1 + (-8.51 + 26.2i)T + (-1.07e3 - 782. i)T^{2} \) |
| 13 | \( 1 + (-10.5 - 32.3i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (50.5 - 36.7i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (33.5 - 24.3i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-56.6 + 174. i)T + (-9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (9.59 + 6.97i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (45.2 - 32.9i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-92.5 - 284. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (15.1 + 46.6i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 377.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-28.4 - 20.6i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-546. - 397. i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-17.1 - 52.7i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-58.2 + 179. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + (82.9 - 60.2i)T + (9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (860. + 625. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (189. - 584. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-439. - 319. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (672. - 488. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (-44.4 + 136. i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (763. + 554. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−16.90893795713059959219660158588, −16.34856719359360023960603057564, −15.23204605214609889758500706560, −14.26981775812210370571548861195, −13.31483063685166948880156707987, −10.07434647713401219463992364692, −8.871030189517515728815922387304, −8.484215737798618342346311995696, −6.34914569169278031593413199494, −4.50949740810756569442431981304,
1.99446924357686567740498279264, 3.25744978954984348023526343540, 7.31187380651330805403687352675, 8.915814164485210915832559107431, 9.944669128488005343294615339679, 11.37270802669406066811010557077, 12.96187884919182109148536804214, 13.50812976844076688402464307925, 14.91986624822805263375853363464, 17.65131198755523948968195414952
