| L(s) = 1 | + (−7.08 + 1.89i)2-s + (−0.547 + 0.316i)3-s + (32.7 − 18.8i)4-s + (12.4 + 3.33i)5-s + (3.27 − 3.27i)6-s + (−4.56 − 7.90i)7-s + (−112. + 112. i)8-s + (−40.2 + 69.8i)9-s − 94.4·10-s + 111. i·11-s + (−11.9 + 20.6i)12-s + (−107. − 28.6i)13-s + (47.3 + 47.3i)14-s + (−7.86 + 2.10i)15-s + (283. − 490. i)16-s + (104. + 390. i)17-s + ⋯ |
| L(s) = 1 | + (−1.77 + 0.474i)2-s + (−0.0608 + 0.0351i)3-s + (2.04 − 1.18i)4-s + (0.497 + 0.133i)5-s + (0.0910 − 0.0910i)6-s + (−0.0931 − 0.161i)7-s + (−1.76 + 1.76i)8-s + (−0.497 + 0.861i)9-s − 0.944·10-s + 0.922i·11-s + (−0.0829 + 0.143i)12-s + (−0.633 − 0.169i)13-s + (0.241 + 0.241i)14-s + (−0.0349 + 0.00937i)15-s + (1.10 − 1.91i)16-s + (0.361 + 1.34i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.231523 + 0.433072i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.231523 + 0.433072i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (1.32e3 - 355. i)T \) |
| good | 2 | \( 1 + (7.08 - 1.89i)T + (13.8 - 8i)T^{2} \) |
| 3 | \( 1 + (0.547 - 0.316i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (-12.4 - 3.33i)T + (541. + 312.5i)T^{2} \) |
| 7 | \( 1 + (4.56 + 7.90i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 - 111. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (107. + 28.6i)T + (2.47e4 + 1.42e4i)T^{2} \) |
| 17 | \( 1 + (-104. - 390. i)T + (-7.23e4 + 4.17e4i)T^{2} \) |
| 19 | \( 1 + (-228. - 61.3i)T + (1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (595. - 595. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 + (-24.0 - 24.0i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + (-1.19e3 - 1.19e3i)T + 9.23e5iT^{2} \) |
| 41 | \( 1 + (-1.81e3 + 1.04e3i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-1.48e3 + 1.48e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + 2.36e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (-1.36e3 + 2.37e3i)T + (-3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-505. - 1.88e3i)T + (-1.04e7 + 6.05e6i)T^{2} \) |
| 61 | \( 1 + (829. - 3.09e3i)T + (-1.19e7 - 6.92e6i)T^{2} \) |
| 67 | \( 1 + (2.71e3 - 1.56e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (867. + 1.50e3i)T + (-1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 - 229. iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (-8.50e3 - 2.28e3i)T + (3.37e7 + 1.94e7i)T^{2} \) |
| 83 | \( 1 + (-999. + 1.73e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (1.35e4 - 3.63e3i)T + (5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (-5.13e3 + 5.13e3i)T - 8.85e7iT^{2} \) |
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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
−16.36930466737530321748369798145, −15.31035212830642708149468019753, −13.99942850782855105541725972461, −11.99263201551416748028940649949, −10.41630049934276098663368439023, −9.878874076955926961382888484665, −8.339218650099687070308106924220, −7.29200900698580235221904263696, −5.74546198723044835688418079448, −1.93282277262776408334811017754,
0.58007531583719755504260723642, 2.73748788906698400589101566510, 6.19991752069690788363085001376, 7.80349445197149283982083979504, 9.144425317609015978669895736286, 9.841857077637210607728396407436, 11.35255781744990619917619027745, 12.16655702229912044310311738291, 14.06556660677970499989900775216, 15.81277931362233660517530425625
