| 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 | \( 1 + (117. - 740. i)T + (-8.68e5 - 2.82e5i)T^{2} \) |
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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
−12.26043869343087201004044650145, −10.88804388071032081765900903986, −9.693798670182584810641339454222, −9.035098700377352918420485961649, −7.990133013576499261913267217447, −7.01272861939209785622400755531, −6.42071411737072304896055561635, −3.80034789607156704017049658768, −2.50414113038047009225451903730, −0.875459045095436970796706456436,
1.28925142385483709197359397905, 3.63747474713698039572690684837, 4.30003833590056142548016369068, 5.99875089517887109492149830163, 7.83609116353689286760033135070, 8.916911822469795900650881186438, 9.225516763349434696172849658218, 9.867406968861939895246711865411, 11.01640454221087448566761613962, 12.45656141216963880305355408351
