| L(s) = 1 | + (0.143 − 0.00750i)2-s + (−0.396 + 0.489i)3-s + (−7.93 + 0.834i)4-s + (−6.20 − 9.30i)5-s + (−0.0531 + 0.0731i)6-s + (−18.1 + 3.69i)7-s + (−2.26 + 0.358i)8-s + (5.53 + 26.0i)9-s + (−0.958 − 1.28i)10-s + (62.9 + 13.3i)11-s + (2.73 − 4.21i)12-s + (50.0 − 25.5i)13-s + (−2.57 + 0.665i)14-s + (7.01 + 0.649i)15-s + (62.1 − 13.2i)16-s + (7.31 − 19.0i)17-s + ⋯ |
| L(s) = 1 | + (0.0506 − 0.00265i)2-s + (−0.0763 + 0.0942i)3-s + (−0.991 + 0.104i)4-s + (−0.555 − 0.831i)5-s + (−0.00361 + 0.00497i)6-s + (−0.979 + 0.199i)7-s + (−0.100 + 0.0158i)8-s + (0.204 + 0.963i)9-s + (−0.0303 − 0.0406i)10-s + (1.72 + 0.366i)11-s + (0.0659 − 0.101i)12-s + (1.06 − 0.544i)13-s + (−0.0490 + 0.0126i)14-s + (0.120 + 0.0111i)15-s + (0.970 − 0.206i)16-s + (0.104 − 0.272i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.204i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.16563 + 0.120336i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16563 + 0.120336i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (6.20 + 9.30i)T \) |
| 7 | \( 1 + (18.1 - 3.69i)T \) |
| good | 2 | \( 1 + (-0.143 + 0.00750i)T + (7.95 - 0.836i)T^{2} \) |
| 3 | \( 1 + (0.396 - 0.489i)T + (-5.61 - 26.4i)T^{2} \) |
| 11 | \( 1 + (-62.9 - 13.3i)T + (1.21e3 + 541. i)T^{2} \) |
| 13 | \( 1 + (-50.0 + 25.5i)T + (1.29e3 - 1.77e3i)T^{2} \) |
| 17 | \( 1 + (-7.31 + 19.0i)T + (-3.65e3 - 3.28e3i)T^{2} \) |
| 19 | \( 1 + (-7.32 + 69.7i)T + (-6.70e3 - 1.42e3i)T^{2} \) |
| 23 | \( 1 + (-2.91 - 55.6i)T + (-1.21e4 + 1.27e3i)T^{2} \) |
| 29 | \( 1 + (-56.0 - 77.1i)T + (-7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-30.4 - 68.3i)T + (-1.99e4 + 2.21e4i)T^{2} \) |
| 37 | \( 1 + (-126. - 82.1i)T + (2.06e4 + 4.62e4i)T^{2} \) |
| 41 | \( 1 + (214. + 69.7i)T + (5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + (-269. + 269. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (285. - 109. i)T + (7.71e4 - 6.94e4i)T^{2} \) |
| 53 | \( 1 + (-326. - 264. i)T + (3.09e4 + 1.45e5i)T^{2} \) |
| 59 | \( 1 + (-420. - 467. i)T + (-2.14e4 + 2.04e5i)T^{2} \) |
| 61 | \( 1 + (-481. - 433. i)T + (2.37e4 + 2.25e5i)T^{2} \) |
| 67 | \( 1 + (-1.63 - 0.625i)T + (2.23e5 + 2.01e5i)T^{2} \) |
| 71 | \( 1 + (-663. + 481. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-140. - 216. i)T + (-1.58e5 + 3.55e5i)T^{2} \) |
| 79 | \( 1 + (250. - 563. i)T + (-3.29e5 - 3.66e5i)T^{2} \) |
| 83 | \( 1 + (99.9 + 631. i)T + (-5.43e5 + 1.76e5i)T^{2} \) |
| 89 | \( 1 + (-628. + 698. i)T + (-7.36e4 - 7.01e5i)T^{2} \) |
| 97 | \( 1 + (-170. + 1.07e3i)T + (-8.68e5 - 2.82e5i)T^{2} \) |
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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
−12.43428393576851553058707797678, −11.48798809535879775305268476032, −10.07137561793641155075542824022, −9.093123101646588272281242200211, −8.521586191601331034402362795963, −7.12753372436396343388966611264, −5.64573169344758219950288774736, −4.48134931710315621304914687644, −3.53402369829251951271696353131, −0.966812023803598229441343558297,
0.826197120608440299810110480705, 3.69326057012525272178296949364, 3.87858692802877991054756450103, 6.22872861946262901394067377761, 6.59393675495045044956350745064, 8.271763594143884744991386007688, 9.293372682881540912899519786458, 10.01979915853730969964323953226, 11.37246432104077993084016054888, 12.20766128862003190414345602975
