| L(s) = 1 | + (−4.23 − 0.221i)2-s + (−3.57 − 4.41i)3-s + (9.93 + 1.04i)4-s + (11.1 + 0.705i)5-s + (14.1 + 19.4i)6-s + (17.1 + 7.03i)7-s + (−8.34 − 1.32i)8-s + (−1.09 + 5.14i)9-s + (−47.1 − 5.46i)10-s + (−65.1 + 13.8i)11-s + (−30.9 − 47.5i)12-s + (−2.69 − 1.37i)13-s + (−71.0 − 33.6i)14-s + (−36.7 − 51.7i)15-s + (−43.1 − 9.16i)16-s + (49.9 + 130. i)17-s + ⋯ |
| L(s) = 1 | + (−1.49 − 0.0784i)2-s + (−0.687 − 0.849i)3-s + (1.24 + 0.130i)4-s + (0.998 + 0.0630i)5-s + (0.963 + 1.32i)6-s + (0.925 + 0.379i)7-s + (−0.368 − 0.0584i)8-s + (−0.0404 + 0.190i)9-s + (−1.48 − 0.172i)10-s + (−1.78 + 0.379i)11-s + (−0.743 − 1.14i)12-s + (−0.0575 − 0.0293i)13-s + (−1.35 − 0.641i)14-s + (−0.632 − 0.891i)15-s + (−0.673 − 0.143i)16-s + (0.712 + 1.85i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.573966 + 0.195261i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.573966 + 0.195261i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-11.1 - 0.705i)T \) |
| 7 | \( 1 + (-17.1 - 7.03i)T \) |
| good | 2 | \( 1 + (4.23 + 0.221i)T + (7.95 + 0.836i)T^{2} \) |
| 3 | \( 1 + (3.57 + 4.41i)T + (-5.61 + 26.4i)T^{2} \) |
| 11 | \( 1 + (65.1 - 13.8i)T + (1.21e3 - 541. i)T^{2} \) |
| 13 | \( 1 + (2.69 + 1.37i)T + (1.29e3 + 1.77e3i)T^{2} \) |
| 17 | \( 1 + (-49.9 - 130. i)T + (-3.65e3 + 3.28e3i)T^{2} \) |
| 19 | \( 1 + (-3.14 - 29.9i)T + (-6.70e3 + 1.42e3i)T^{2} \) |
| 23 | \( 1 + (-3.34 + 63.8i)T + (-1.21e4 - 1.27e3i)T^{2} \) |
| 29 | \( 1 + (80.3 - 110. i)T + (-7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (54.6 - 122. i)T + (-1.99e4 - 2.21e4i)T^{2} \) |
| 37 | \( 1 + (-67.4 + 43.7i)T + (2.06e4 - 4.62e4i)T^{2} \) |
| 41 | \( 1 + (-157. + 51.1i)T + (5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + (-52.0 - 52.0i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (-267. - 102. i)T + (7.71e4 + 6.94e4i)T^{2} \) |
| 53 | \( 1 + (499. - 404. i)T + (3.09e4 - 1.45e5i)T^{2} \) |
| 59 | \( 1 + (558. - 620. i)T + (-2.14e4 - 2.04e5i)T^{2} \) |
| 61 | \( 1 + (-557. + 502. i)T + (2.37e4 - 2.25e5i)T^{2} \) |
| 67 | \( 1 + (-698. + 267. i)T + (2.23e5 - 2.01e5i)T^{2} \) |
| 71 | \( 1 + (-248. - 180. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-360. + 555. i)T + (-1.58e5 - 3.55e5i)T^{2} \) |
| 79 | \( 1 + (-101. - 228. i)T + (-3.29e5 + 3.66e5i)T^{2} \) |
| 83 | \( 1 + (-78.0 + 492. i)T + (-5.43e5 - 1.76e5i)T^{2} \) |
| 89 | \( 1 + (-377. - 418. i)T + (-7.36e4 + 7.01e5i)T^{2} \) |
| 97 | \( 1 + (-162. - 1.02e3i)T + (-8.68e5 + 2.82e5i)T^{2} \) |
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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
−12.37628694416241470685469434745, −10.76460717162131638566342325356, −10.60865819013033033106182012931, −9.327507681892167060814463366203, −8.167862948939658397168156907533, −7.50989212482119689751854217780, −6.18153574850963052580183672025, −5.21503192415873663035904075374, −2.20948018136201745352679186799, −1.27939372986389377334997821314,
0.53668226322172993228003092802, 2.33134835393061587407517604533, 4.86727097025416770603215244952, 5.54953959139182688569171464257, 7.33972276844449071330433969908, 8.103114279917202954054357866675, 9.518370233741221707922436139147, 9.959898965763092961500121620501, 10.92578153237046031411559900149, 11.35645698050762031672136719657
