| L(s) = 1 | + (1.20 − 0.874i)2-s + (−0.927 + 2.85i)3-s + (−1.78 + 5.50i)4-s + (−10.4 − 4.01i)5-s + (1.37 + 4.24i)6-s − 27.0·7-s + (6.33 + 19.5i)8-s + (−7.28 − 5.29i)9-s + (−16.0 + 4.29i)10-s + (10.1 − 7.35i)11-s + (−14.0 − 10.2i)12-s + (−7.47 − 5.43i)13-s + (−32.5 + 23.6i)14-s + (21.1 − 26.0i)15-s + (−12.7 − 9.26i)16-s + (33.8 + 104. i)17-s + ⋯ |
| L(s) = 1 | + (0.425 − 0.309i)2-s + (−0.178 + 0.549i)3-s + (−0.223 + 0.687i)4-s + (−0.933 − 0.358i)5-s + (0.0938 + 0.288i)6-s − 1.46·7-s + (0.280 + 0.862i)8-s + (−0.269 − 0.195i)9-s + (−0.508 + 0.135i)10-s + (0.277 − 0.201i)11-s + (−0.337 − 0.245i)12-s + (−0.159 − 0.115i)13-s + (−0.621 + 0.451i)14-s + (0.363 − 0.448i)15-s + (−0.199 − 0.144i)16-s + (0.483 + 1.48i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 - 0.571i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.189513 + 0.603842i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.189513 + 0.603842i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.927 - 2.85i)T \) |
| 5 | \( 1 + (10.4 + 4.01i)T \) |
| good | 2 | \( 1 + (-1.20 + 0.874i)T + (2.47 - 7.60i)T^{2} \) |
| 7 | \( 1 + 27.0T + 343T^{2} \) |
| 11 | \( 1 + (-10.1 + 7.35i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (7.47 + 5.43i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-33.8 - 104. i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-27.7 - 85.4i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (17.7 - 12.9i)T + (3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (4.78 - 14.7i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (87.3 + 268. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (182. + 132. i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-232. - 168. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 285.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-10.2 + 31.5i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (219. - 676. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-159. - 115. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-195. + 142. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-131. - 403. i)T + (-2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-134. + 412. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (929. - 675. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (299. - 920. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (154. + 476. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (-364. + 264. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-467. + 1.43e3i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−14.46781768891789800448334500338, −12.98944626533704169241913625920, −12.44567958884468280124320085591, −11.43490187978501603523597039467, −10.07427548097984347036178169079, −8.812206301280980283937427515372, −7.64264738395567601790387397803, −5.86630844303714530927003640418, −4.08059584690147269547568869126, −3.37770599548360995791278944528,
0.35727014780066438708941441317, 3.27935780988327869093231008865, 5.02139188568291168058951344061, 6.61479669552235310479911453636, 7.17392670397050498815850929304, 9.120070083521847868335151263319, 10.22264236279045562597909162007, 11.61223162641530185416987703733, 12.63911540842940417461904347892, 13.65355763277855154177550705695
