| L(s) = 1 | + (0.147 − 2.06i)2-s + (5.23 − 1.13i)3-s + (3.67 + 0.528i)4-s + (4.29 + 10.3i)5-s + (−1.57 − 10.9i)6-s + (3.96 − 2.16i)7-s + (5.15 − 23.6i)8-s + (1.57 − 0.719i)9-s + (21.9 − 7.33i)10-s + (42.7 + 37.0i)11-s + (19.8 − 1.41i)12-s + (−21.9 + 40.1i)13-s + (−3.89 − 8.51i)14-s + (34.2 + 49.1i)15-s + (−19.7 − 5.78i)16-s + (−90.8 − 67.9i)17-s + ⋯ |
| L(s) = 1 | + (0.0522 − 0.730i)2-s + (1.00 − 0.219i)3-s + (0.459 + 0.0660i)4-s + (0.383 + 0.923i)5-s + (−0.107 − 0.747i)6-s + (0.214 − 0.117i)7-s + (0.227 − 1.04i)8-s + (0.0583 − 0.0266i)9-s + (0.694 − 0.232i)10-s + (1.17 + 1.01i)11-s + (0.477 − 0.0341i)12-s + (−0.468 + 0.857i)13-s + (−0.0742 − 0.162i)14-s + (0.589 + 0.846i)15-s + (−0.307 − 0.0904i)16-s + (−1.29 − 0.969i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.598i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.57978 - 0.856496i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.57978 - 0.856496i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-4.29 - 10.3i)T \) |
| 23 | \( 1 + (-20.3 + 108. i)T \) |
| good | 2 | \( 1 + (-0.147 + 2.06i)T + (-7.91 - 1.13i)T^{2} \) |
| 3 | \( 1 + (-5.23 + 1.13i)T + (24.5 - 11.2i)T^{2} \) |
| 7 | \( 1 + (-3.96 + 2.16i)T + (185. - 288. i)T^{2} \) |
| 11 | \( 1 + (-42.7 - 37.0i)T + (189. + 1.31e3i)T^{2} \) |
| 13 | \( 1 + (21.9 - 40.1i)T + (-1.18e3 - 1.84e3i)T^{2} \) |
| 17 | \( 1 + (90.8 + 67.9i)T + (1.38e3 + 4.71e3i)T^{2} \) |
| 19 | \( 1 + (-16.5 + 115. i)T + (-6.58e3 - 1.93e3i)T^{2} \) |
| 29 | \( 1 + (67.4 - 9.69i)T + (2.34e4 - 6.87e3i)T^{2} \) |
| 31 | \( 1 + (-88.1 + 56.6i)T + (1.23e4 - 2.70e4i)T^{2} \) |
| 37 | \( 1 + (231. + 86.3i)T + (3.82e4 + 3.31e4i)T^{2} \) |
| 41 | \( 1 + (-15.1 + 33.1i)T + (-4.51e4 - 5.20e4i)T^{2} \) |
| 43 | \( 1 + (-31.9 - 146. i)T + (-7.23e4 + 3.30e4i)T^{2} \) |
| 47 | \( 1 + (68.7 + 68.7i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 + (139. + 254. i)T + (-8.04e4 + 1.25e5i)T^{2} \) |
| 59 | \( 1 + (-226. - 770. i)T + (-1.72e5 + 1.11e5i)T^{2} \) |
| 61 | \( 1 + (-262. - 408. i)T + (-9.42e4 + 2.06e5i)T^{2} \) |
| 67 | \( 1 + (-583. - 41.6i)T + (2.97e5 + 4.28e4i)T^{2} \) |
| 71 | \( 1 + (45.9 + 52.9i)T + (-5.09e4 + 3.54e5i)T^{2} \) |
| 73 | \( 1 + (-523. - 698. i)T + (-1.09e5 + 3.73e5i)T^{2} \) |
| 79 | \( 1 + (363. - 106. i)T + (4.14e5 - 2.66e5i)T^{2} \) |
| 83 | \( 1 + (-299. + 803. i)T + (-4.32e5 - 3.74e5i)T^{2} \) |
| 89 | \( 1 + (1.21e3 + 782. i)T + (2.92e5 + 6.41e5i)T^{2} \) |
| 97 | \( 1 + (-201. - 541. i)T + (-6.89e5 + 5.97e5i)T^{2} \) |
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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
−13.06249053086333445466135080309, −11.70423702867284897121539615446, −11.11240069405337539823280299076, −9.754414999972209773149772065614, −8.998487915909067345919753573231, −7.12003431959566789426657122765, −6.82812788652737872495960962111, −4.31168269505984735798369551443, −2.75200490026345416001338772775, −2.00820489188341943880733099209,
1.81677335194133437290050214062, 3.61294713662993471430958689559, 5.39108141576135527374811937517, 6.39108394743070254120396193127, 8.097329822624443177684600449117, 8.540585860429130489064167947528, 9.669505457518163860869912827295, 11.13750059350398008246742075668, 12.26272745072083238552496014391, 13.59586470045779558132775455618
