| L(s) = 1 | + (−2.63 − 0.137i)2-s + (4.95 + 6.11i)3-s + (−1.04 − 0.110i)4-s + (0.357 − 11.1i)5-s + (−12.1 − 16.7i)6-s + (−18.4 − 0.896i)7-s + (23.5 + 3.73i)8-s + (−7.25 + 34.1i)9-s + (−2.48 + 29.3i)10-s + (53.7 − 11.4i)11-s + (−4.52 − 6.96i)12-s + (34.3 + 17.5i)13-s + (48.5 + 4.91i)14-s + (70.0 − 53.1i)15-s + (−53.2 − 11.3i)16-s + (25.2 + 65.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.930 − 0.0487i)2-s + (0.952 + 1.17i)3-s + (−0.131 − 0.0137i)4-s + (0.0319 − 0.999i)5-s + (−0.829 − 1.14i)6-s + (−0.998 − 0.0484i)7-s + (1.04 + 0.164i)8-s + (−0.268 + 1.26i)9-s + (−0.0784 + 0.928i)10-s + (1.47 − 0.312i)11-s + (−0.108 − 0.167i)12-s + (0.733 + 0.373i)13-s + (0.926 + 0.0937i)14-s + (1.20 − 0.914i)15-s + (−0.832 − 0.176i)16-s + (0.359 + 0.936i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 - 0.872i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.489 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.09478 + 0.640969i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09478 + 0.640969i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.357 + 11.1i)T \) |
| 7 | \( 1 + (18.4 + 0.896i)T \) |
| good | 2 | \( 1 + (2.63 + 0.137i)T + (7.95 + 0.836i)T^{2} \) |
| 3 | \( 1 + (-4.95 - 6.11i)T + (-5.61 + 26.4i)T^{2} \) |
| 11 | \( 1 + (-53.7 + 11.4i)T + (1.21e3 - 541. i)T^{2} \) |
| 13 | \( 1 + (-34.3 - 17.5i)T + (1.29e3 + 1.77e3i)T^{2} \) |
| 17 | \( 1 + (-25.2 - 65.6i)T + (-3.65e3 + 3.28e3i)T^{2} \) |
| 19 | \( 1 + (-1.12 - 10.6i)T + (-6.70e3 + 1.42e3i)T^{2} \) |
| 23 | \( 1 + (9.13 - 174. i)T + (-1.21e4 - 1.27e3i)T^{2} \) |
| 29 | \( 1 + (-72.7 + 100. i)T + (-7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (72.1 - 161. i)T + (-1.99e4 - 2.21e4i)T^{2} \) |
| 37 | \( 1 + (-145. + 94.5i)T + (2.06e4 - 4.62e4i)T^{2} \) |
| 41 | \( 1 + (-228. + 74.2i)T + (5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + (-259. - 259. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (-150. - 57.8i)T + (7.71e4 + 6.94e4i)T^{2} \) |
| 53 | \( 1 + (361. - 292. i)T + (3.09e4 - 1.45e5i)T^{2} \) |
| 59 | \( 1 + (-104. + 116. i)T + (-2.14e4 - 2.04e5i)T^{2} \) |
| 61 | \( 1 + (667. - 601. i)T + (2.37e4 - 2.25e5i)T^{2} \) |
| 67 | \( 1 + (-935. + 358. i)T + (2.23e5 - 2.01e5i)T^{2} \) |
| 71 | \( 1 + (392. + 285. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-451. + 695. i)T + (-1.58e5 - 3.55e5i)T^{2} \) |
| 79 | \( 1 + (-332. - 746. i)T + (-3.29e5 + 3.66e5i)T^{2} \) |
| 83 | \( 1 + (-121. + 768. i)T + (-5.43e5 - 1.76e5i)T^{2} \) |
| 89 | \( 1 + (323. + 359. i)T + (-7.36e4 + 7.01e5i)T^{2} \) |
| 97 | \( 1 + (117. + 740. i)T + (-8.68e5 + 2.82e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.45656141216963880305355408351, −11.01640454221087448566761613962, −9.867406968861939895246711865411, −9.225516763349434696172849658218, −8.916911822469795900650881186438, −7.83609116353689286760033135070, −5.99875089517887109492149830163, −4.30003833590056142548016369068, −3.63747474713698039572690684837, −1.28925142385483709197359397905,
0.875459045095436970796706456436, 2.50414113038047009225451903730, 3.80034789607156704017049658768, 6.42071411737072304896055561635, 7.01272861939209785622400755531, 7.990133013576499261913267217447, 9.035098700377352918420485961649, 9.693798670182584810641339454222, 10.88804388071032081765900903986, 12.26043869343087201004044650145
