| L(s) = 1 | + (−4.93 + 1.12i)2-s + (8.18 + 1.86i)3-s + (15.8 − 7.62i)4-s + (3.18 − 6.62i)5-s − 42.4·6-s + (1.15 − 5.04i)7-s + (−37.8 + 30.2i)8-s + (39.1 + 18.8i)9-s + (−8.27 + 36.2i)10-s + (25.8 − 5.91i)11-s + (143. − 32.8i)12-s + (23.1 + 18.4i)13-s + 26.1i·14-s + (38.4 − 48.2i)15-s + (65.1 − 81.6i)16-s + (7.25 − 15.0i)17-s + ⋯ |
| L(s) = 1 | + (−1.74 + 0.397i)2-s + (1.57 + 0.359i)3-s + (1.97 − 0.953i)4-s + (0.285 − 0.592i)5-s − 2.88·6-s + (0.0621 − 0.272i)7-s + (−1.67 + 1.33i)8-s + (1.45 + 0.698i)9-s + (−0.261 + 1.14i)10-s + (0.709 − 0.162i)11-s + (3.46 − 0.789i)12-s + (0.494 + 0.394i)13-s + 0.499i·14-s + (0.662 − 0.830i)15-s + (1.01 − 1.27i)16-s + (0.103 − 0.214i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.121i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.992 - 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.49776 + 0.0912270i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49776 + 0.0912270i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + (-1.20e3 + 2.48e3i)T \) |
| good | 2 | \( 1 + (4.93 - 1.12i)T + (7.20 - 3.47i)T^{2} \) |
| 3 | \( 1 + (-8.18 - 1.86i)T + (24.3 + 11.7i)T^{2} \) |
| 5 | \( 1 + (-3.18 + 6.62i)T + (-77.9 - 97.7i)T^{2} \) |
| 7 | \( 1 + (-1.15 + 5.04i)T + (-309. - 148. i)T^{2} \) |
| 11 | \( 1 + (-25.8 + 5.91i)T + (1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (-23.1 - 18.4i)T + (488. + 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-7.25 + 15.0i)T + (-3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 + 31.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + (32.9 + 144. i)T + (-1.09e4 + 5.27e3i)T^{2} \) |
| 29 | \( 1 + (29.7 + 130. i)T + (-2.19e4 + 1.05e4i)T^{2} \) |
| 31 | \( 1 + (-159. + 36.4i)T + (2.68e4 - 1.29e4i)T^{2} \) |
| 37 | \( 1 + (-221. - 278. i)T + (-1.12e4 + 4.93e4i)T^{2} \) |
| 41 | \( 1 + (143. + 69.0i)T + (4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (98.5 + 431. i)T + (-7.16e4 + 3.44e4i)T^{2} \) |
| 47 | \( 1 + (-219. - 274. i)T + (-2.31e4 + 1.01e5i)T^{2} \) |
| 53 | \( 1 + (68.0 - 32.7i)T + (9.28e4 - 1.16e5i)T^{2} \) |
| 59 | \( 1 + (33.1 + 145. i)T + (-1.85e5 + 8.91e4i)T^{2} \) |
| 61 | \( 1 + (-157. - 691. i)T + (-2.04e5 + 9.84e4i)T^{2} \) |
| 67 | \( 1 + (-150. + 119. i)T + (6.69e4 - 2.93e5i)T^{2} \) |
| 71 | \( 1 + (437. - 907. i)T + (-2.23e5 - 2.79e5i)T^{2} \) |
| 73 | \( 1 + (210. - 167. i)T + (8.65e4 - 3.79e5i)T^{2} \) |
| 79 | \( 1 + (247. + 514. i)T + (-3.07e5 + 3.85e5i)T^{2} \) |
| 83 | \( 1 - 514.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (485. + 110. i)T + (6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 + (227. + 109. i)T + (5.69e5 + 7.13e5i)T^{2} \) |
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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
−11.72454668741035447229818321789, −10.43097515896401252467534085293, −9.699999301225574096509998717335, −8.795507384635870428050639806527, −8.524686411977234450587784809591, −7.44438836776553567463246363619, −6.30204523362256341060305685467, −4.20639443014894254396601111054, −2.46674325980225438445175787331, −1.14784043162500858584377555147,
1.40539188849730914082470022787, 2.46182118980919778434657154491, 3.50440472035969963347307938561, 6.42677085592603641620651229632, 7.43639556328173620258730094000, 8.241231784020657768672553746337, 9.007269124703987559904544977609, 9.719660793677155453050109780603, 10.64563161655910059596888560471, 11.73973628920147073891146824845
