| L(s) = 1 | + (1.13 + 0.0597i)2-s + (−4.65 − 5.74i)3-s + (−6.66 − 0.700i)4-s + (−0.689 − 11.1i)5-s + (−4.95 − 6.81i)6-s + (−17.8 + 4.85i)7-s + (−16.5 − 2.62i)8-s + (−5.73 + 26.9i)9-s + (−0.118 − 12.7i)10-s + (9.78 − 2.07i)11-s + (26.9 + 41.5i)12-s + (62.8 + 32.0i)13-s + (−20.6 + 4.46i)14-s + (−60.8 + 55.8i)15-s + (33.7 + 7.16i)16-s + (9.43 + 24.5i)17-s + ⋯ |
| L(s) = 1 | + (0.402 + 0.0211i)2-s + (−0.894 − 1.10i)3-s + (−0.832 − 0.0875i)4-s + (−0.0616 − 0.998i)5-s + (−0.337 − 0.464i)6-s + (−0.965 + 0.262i)7-s + (−0.731 − 0.115i)8-s + (−0.212 + 0.999i)9-s + (−0.00375 − 0.403i)10-s + (0.268 − 0.0569i)11-s + (0.648 + 0.998i)12-s + (1.34 + 0.682i)13-s + (−0.394 + 0.0852i)14-s + (−1.04 + 0.961i)15-s + (0.526 + 0.111i)16-s + (0.134 + 0.350i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0103 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0103 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0148279 + 0.0146747i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0148279 + 0.0146747i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.689 + 11.1i)T \) |
| 7 | \( 1 + (17.8 - 4.85i)T \) |
| good | 2 | \( 1 + (-1.13 - 0.0597i)T + (7.95 + 0.836i)T^{2} \) |
| 3 | \( 1 + (4.65 + 5.74i)T + (-5.61 + 26.4i)T^{2} \) |
| 11 | \( 1 + (-9.78 + 2.07i)T + (1.21e3 - 541. i)T^{2} \) |
| 13 | \( 1 + (-62.8 - 32.0i)T + (1.29e3 + 1.77e3i)T^{2} \) |
| 17 | \( 1 + (-9.43 - 24.5i)T + (-3.65e3 + 3.28e3i)T^{2} \) |
| 19 | \( 1 + (9.26 + 88.1i)T + (-6.70e3 + 1.42e3i)T^{2} \) |
| 23 | \( 1 + (-6.88 + 131. i)T + (-1.21e4 - 1.27e3i)T^{2} \) |
| 29 | \( 1 + (182. - 250. i)T + (-7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (97.0 - 218. i)T + (-1.99e4 - 2.21e4i)T^{2} \) |
| 37 | \( 1 + (-20.9 + 13.6i)T + (2.06e4 - 4.62e4i)T^{2} \) |
| 41 | \( 1 + (201. - 65.6i)T + (5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + (370. + 370. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (-496. - 190. i)T + (7.71e4 + 6.94e4i)T^{2} \) |
| 53 | \( 1 + (63.1 - 51.1i)T + (3.09e4 - 1.45e5i)T^{2} \) |
| 59 | \( 1 + (169. - 188. i)T + (-2.14e4 - 2.04e5i)T^{2} \) |
| 61 | \( 1 + (388. - 349. i)T + (2.37e4 - 2.25e5i)T^{2} \) |
| 67 | \( 1 + (37.4 - 14.3i)T + (2.23e5 - 2.01e5i)T^{2} \) |
| 71 | \( 1 + (-161. - 117. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-406. + 625. i)T + (-1.58e5 - 3.55e5i)T^{2} \) |
| 79 | \( 1 + (-23.1 - 51.9i)T + (-3.29e5 + 3.66e5i)T^{2} \) |
| 83 | \( 1 + (92.2 - 582. i)T + (-5.43e5 - 1.76e5i)T^{2} \) |
| 89 | \( 1 + (247. + 275. i)T + (-7.36e4 + 7.01e5i)T^{2} \) |
| 97 | \( 1 + (18.5 + 116. i)T + (-8.68e5 + 2.82e5i)T^{2} \) |
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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
−12.60795802208276155530759942668, −12.01104147877592182430295299428, −10.74179191052972269358683855103, −9.093973573079360476171658190053, −8.703922048881203356040253558513, −6.92109699855181882202049364916, −6.06015829332649472268522066161, −5.12316827637382785219259340653, −3.72433515379229636449263162692, −1.27545021796703221015024229010,
0.01071997214894981712101205303, 3.50126331129900013691467979910, 3.94673058613932491008978915841, 5.62386133622373689622189843368, 6.15789831120040066765240674881, 7.84849818156488234339233940166, 9.496600985137906484304546830716, 9.968476093295875716294243555580, 10.99259068807458548240129975249, 11.81005958850736730329337193322
