| L(s) = 1 | + (−0.217 + 0.299i)2-s + (2.64 − 0.859i)3-s + (2.42 + 7.47i)4-s + (7.78 − 8.02i)5-s + (−0.317 + 0.978i)6-s + 0.707i·7-s + (−5.58 − 1.81i)8-s + (−15.5 + 11.3i)9-s + (0.708 + 4.07i)10-s + (−36.2 − 26.3i)11-s + (12.8 + 17.6i)12-s + (−28.2 − 38.8i)13-s + (−0.211 − 0.153i)14-s + (13.6 − 27.9i)15-s + (−49.1 + 35.6i)16-s + (95.0 + 30.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.0769 + 0.105i)2-s + (0.508 − 0.165i)3-s + (0.303 + 0.934i)4-s + (0.696 − 0.717i)5-s + (−0.0216 + 0.0665i)6-s + 0.0382i·7-s + (−0.246 − 0.0801i)8-s + (−0.577 + 0.419i)9-s + (0.0223 + 0.128i)10-s + (−0.993 − 0.721i)11-s + (0.309 + 0.425i)12-s + (−0.601 − 0.828i)13-s + (−0.00404 − 0.00293i)14-s + (0.235 − 0.480i)15-s + (−0.767 + 0.557i)16-s + (1.35 + 0.440i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.32798 + 0.145521i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32798 + 0.145521i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-7.78 + 8.02i)T \) |
| good | 2 | \( 1 + (0.217 - 0.299i)T + (-2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (-2.64 + 0.859i)T + (21.8 - 15.8i)T^{2} \) |
| 7 | \( 1 - 0.707iT - 343T^{2} \) |
| 11 | \( 1 + (36.2 + 26.3i)T + (411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (28.2 + 38.8i)T + (-678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-95.0 - 30.8i)T + (3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (17.2 - 52.9i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-68.7 + 94.5i)T + (-3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-35.9 - 110. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (19.5 - 60.2i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (69.5 + 95.6i)T + (-1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-147. + 106. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 418. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (146. - 47.4i)T + (8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-475. + 154. i)T + (1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (74.8 - 54.3i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-522. - 379. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + (890. + 289. i)T + (2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (48.2 + 148. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-353. + 486. i)T + (-1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-208. - 640. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (1.27e3 + 414. i)T + (4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (109. + 79.2i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-277. + 90.3i)T + (7.38e5 - 5.36e5i)T^{2} \) |
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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
−16.93923616622867416370529218050, −16.27936447144993470525021532565, −14.53836039030954378050384803802, −13.19891398609366986547869007513, −12.37977500234990184957266559281, −10.50464225585699137190604754575, −8.659281897009402248782719120335, −7.81152368499671261489424181726, −5.54094854536219008180860370430, −2.84453380039830201696768810603,
2.53097300404160091062810969766, 5.49471063015905721383392336058, 7.14516187544295604394966021915, 9.384042103781677962962784507947, 10.22208970883013968409717794021, 11.67544144463958003873278059435, 13.67499752179922751033756073851, 14.62749888504777106160723368579, 15.41108163180705768682947419623, 17.21631820906101211845734562066
