| L(s) = 1 | + (0.269 − 0.196i)2-s + (−2.60 + 8.02i)3-s + (−2.43 + 7.50i)4-s + (−0.131 − 11.1i)5-s + (0.870 + 2.67i)6-s + 30.0·7-s + (1.63 + 5.04i)8-s + (−35.8 − 26.0i)9-s + (−2.22 − 2.99i)10-s + (−3.55 + 2.58i)11-s + (−53.8 − 39.1i)12-s + (34.4 + 25.0i)13-s + (8.10 − 5.88i)14-s + (90.1 + 28.1i)15-s + (−49.6 − 36.0i)16-s + (−13.2 − 40.8i)17-s + ⋯ |
| L(s) = 1 | + (0.0954 − 0.0693i)2-s + (−0.502 + 1.54i)3-s + (−0.304 + 0.937i)4-s + (−0.0117 − 0.999i)5-s + (0.0592 + 0.182i)6-s + 1.62·7-s + (0.0723 + 0.222i)8-s + (−1.32 − 0.963i)9-s + (−0.0704 − 0.0946i)10-s + (−0.0975 + 0.0708i)11-s + (−1.29 − 0.941i)12-s + (0.734 + 0.533i)13-s + (0.154 − 0.112i)14-s + (1.55 + 0.483i)15-s + (−0.775 − 0.563i)16-s + (−0.189 − 0.582i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.822598 + 0.696995i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.822598 + 0.696995i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.131 + 11.1i)T \) |
| good | 2 | \( 1 + (-0.269 + 0.196i)T + (2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (2.60 - 8.02i)T + (-21.8 - 15.8i)T^{2} \) |
| 7 | \( 1 - 30.0T + 343T^{2} \) |
| 11 | \( 1 + (3.55 - 2.58i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-34.4 - 25.0i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (13.2 + 40.8i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-0.802 - 2.46i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-91.3 + 66.3i)T + (3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-31.2 + 96.2i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (28.6 + 88.1i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-111. - 81.2i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (160. + 116. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 254.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (73.2 - 225. i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-0.435 + 1.34i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-248. - 180. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (401. - 291. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (37.4 + 115. i)T + (-2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-275. + 846. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-90.3 + 65.6i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-165. + 509. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-387. - 1.19e3i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (952. - 692. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-360. + 1.10e3i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−17.03850609439980212639963960952, −16.39679404954582979848901721906, −15.15216347330403494699527228661, −13.64982404547948170814519149995, −11.90675385034570775751570946971, −11.07088460863972659897336218530, −9.236547737303000896993340468497, −8.226170052054206126232984830086, −5.03850509039714973325526487247, −4.25317052517486535082737552666,
1.53720997398985548586709067407, 5.47131540835249473233281458837, 6.81584633057928326080382966510, 8.206828128500853678601384105369, 10.72424776577884928321126663509, 11.47130142940780045504839129021, 13.22395944021066890237984722360, 14.22242998467578992340125262883, 15.15674258120005158372435104170, 17.50564781121160361820720192565
