| L(s) = 1 | + (1.31 − 4.04i)2-s + (13.2 + 9.65i)3-s + (11.2 + 8.15i)4-s + (−5.21 + 55.6i)5-s + (56.5 − 41.0i)6-s − 22.2·7-s + (158. − 114. i)8-s + (8.22 + 25.3i)9-s + (218. + 94.3i)10-s + (94.4 − 290. i)11-s + (70.4 + 216. i)12-s + (−123. − 379. i)13-s + (−29.2 + 90.1i)14-s + (−606. + 688. i)15-s + (−119. − 368. i)16-s + (−1.24e3 + 901. i)17-s + ⋯ |
| L(s) = 1 | + (0.232 − 0.715i)2-s + (0.852 + 0.619i)3-s + (0.350 + 0.254i)4-s + (−0.0933 + 0.995i)5-s + (0.641 − 0.465i)6-s − 0.171·7-s + (0.872 − 0.634i)8-s + (0.0338 + 0.104i)9-s + (0.690 + 0.298i)10-s + (0.235 − 0.724i)11-s + (0.141 + 0.434i)12-s + (−0.202 − 0.623i)13-s + (−0.0399 + 0.122i)14-s + (−0.696 + 0.790i)15-s + (−0.116 − 0.359i)16-s + (−1.04 + 0.756i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0642i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.21956 + 0.0714157i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.21956 + 0.0714157i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (5.21 - 55.6i)T \) |
| good | 2 | \( 1 + (-1.31 + 4.04i)T + (-25.8 - 18.8i)T^{2} \) |
| 3 | \( 1 + (-13.2 - 9.65i)T + (75.0 + 231. i)T^{2} \) |
| 7 | \( 1 + 22.2T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-94.4 + 290. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (123. + 379. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (1.24e3 - 901. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (33.3 - 24.2i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-770. + 2.37e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-5.28e3 - 3.83e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (5.36e3 - 3.90e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (3.50e3 + 1.07e4i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-2.27e3 - 7.01e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 547.T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.83e4 + 1.33e4i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.64e4 - 1.19e4i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-1.40e4 - 4.31e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-4.26e3 + 1.31e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-1.79e3 + 1.30e3i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-6.41e4 - 4.66e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (9.85e3 - 3.03e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-2.82e4 - 2.05e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (4.81e4 - 3.50e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (1.52e3 - 4.68e3i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (7.06e4 + 5.12e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−16.25601332373988930772198784478, −15.15807697562172557726594235209, −14.12698747109075308700814925114, −12.67963564786990136691878805567, −11.12558796041523700612415803594, −10.24066590457210168003677213986, −8.516177932454884181857130404821, −6.73680546990178979947971811654, −3.78232527650580034747880606443, −2.70977293803528232093097753369,
1.91632035066074093036556889542, 4.86183729348481842344567663774, 6.81652561064472729393081220568, 8.020855106586685594128588401226, 9.430728780925350162732587010989, 11.57083958559076941394367443802, 13.12318341313261601932802637687, 14.01936155122112692296727085966, 15.28451438386231398474879328436, 16.31620293036199826017676865756
