| L(s) = 1 | + (−0.451 + 1.34i)2-s + (2.87 + 0.853i)3-s + (−1.59 − 1.21i)4-s + (−5.83 − 2.32i)5-s + (−2.44 + 3.46i)6-s + (6.28 − 2.91i)7-s + (2.34 − 1.58i)8-s + (7.54 + 4.91i)9-s + (5.74 − 6.76i)10-s + (1.90 + 0.527i)11-s + (−3.54 − 4.84i)12-s + (1.67 − 3.16i)13-s + (1.05 + 9.74i)14-s + (−14.7 − 11.6i)15-s + (1.07 + 3.85i)16-s + (4.38 − 9.48i)17-s + ⋯ |
| L(s) = 1 | + (−0.225 + 0.670i)2-s + (0.958 + 0.284i)3-s + (−0.398 − 0.302i)4-s + (−1.16 − 0.464i)5-s + (−0.407 + 0.578i)6-s + (0.898 − 0.415i)7-s + (0.292 − 0.198i)8-s + (0.838 + 0.545i)9-s + (0.574 − 0.676i)10-s + (0.172 + 0.0479i)11-s + (−0.295 − 0.403i)12-s + (0.129 − 0.243i)13-s + (0.0756 + 0.695i)14-s + (−0.986 − 0.777i)15-s + (0.0668 + 0.240i)16-s + (0.258 − 0.557i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.84365 + 0.431567i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.84365 + 0.431567i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.451 - 1.34i)T \) |
| 3 | \( 1 + (-2.87 - 0.853i)T \) |
| 59 | \( 1 + (-54.2 - 23.0i)T \) |
| good | 5 | \( 1 + (5.83 + 2.32i)T + (18.1 + 17.1i)T^{2} \) |
| 7 | \( 1 + (-6.28 + 2.91i)T + (31.7 - 37.3i)T^{2} \) |
| 11 | \( 1 + (-1.90 - 0.527i)T + (103. + 62.3i)T^{2} \) |
| 13 | \( 1 + (-1.67 + 3.16i)T + (-94.8 - 139. i)T^{2} \) |
| 17 | \( 1 + (-4.38 + 9.48i)T + (-187. - 220. i)T^{2} \) |
| 19 | \( 1 + (-14.2 + 3.14i)T + (327. - 151. i)T^{2} \) |
| 23 | \( 1 + (-34.7 - 5.69i)T + (501. + 168. i)T^{2} \) |
| 29 | \( 1 + (0.805 + 2.39i)T + (-669. + 508. i)T^{2} \) |
| 31 | \( 1 + (-48.2 - 10.6i)T + (872. + 403. i)T^{2} \) |
| 37 | \( 1 + (-0.968 + 1.42i)T + (-506. - 1.27e3i)T^{2} \) |
| 41 | \( 1 + (36.6 - 6.01i)T + (1.59e3 - 536. i)T^{2} \) |
| 43 | \( 1 + (13.0 + 46.9i)T + (-1.58e3 + 953. i)T^{2} \) |
| 47 | \( 1 + (-8.14 + 3.24i)T + (1.60e3 - 1.51e3i)T^{2} \) |
| 53 | \( 1 + (35.8 - 30.4i)T + (454. - 2.77e3i)T^{2} \) |
| 61 | \( 1 + (34.3 + 11.5i)T + (2.96e3 + 2.25e3i)T^{2} \) |
| 67 | \( 1 + (8.44 + 12.4i)T + (-1.66e3 + 4.17e3i)T^{2} \) |
| 71 | \( 1 + (-15.0 + 6.00i)T + (3.65e3 - 3.46e3i)T^{2} \) |
| 73 | \( 1 + (121. - 13.2i)T + (5.20e3 - 1.14e3i)T^{2} \) |
| 79 | \( 1 + (-100. + 60.2i)T + (2.92e3 - 5.51e3i)T^{2} \) |
| 83 | \( 1 + (-17.8 + 0.968i)T + (6.84e3 - 744. i)T^{2} \) |
| 89 | \( 1 + (-10.3 - 30.8i)T + (-6.30e3 + 4.79e3i)T^{2} \) |
| 97 | \( 1 + (86.8 + 9.44i)T + (9.18e3 + 2.02e3i)T^{2} \) |
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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
−11.27120988841393708416897087833, −10.22298021063983326216254998950, −9.132264017476677660764419781141, −8.371312686203565059038637200970, −7.70827851060720841191590866575, −7.02156381184690025660167428012, −5.08559840096926165072638809947, −4.41956147467599181057020763265, −3.22328101910564206153621831213, −1.09672089161810795895167320524,
1.31246404119301397142488764788, 2.79713307311297294763162412121, 3.72728692557089781619890447852, 4.82802277556092733685038307882, 6.73444323782811847977903517018, 7.81167075365470573176985596169, 8.291572934644679142787125900763, 9.209048233016331774735644744945, 10.31604878751217923233087077259, 11.40852411403795194517981088157
