| L(s) = 1 | + (0.142 + 1.99i)2-s + (6.09 + 1.32i)3-s + (−3.95 + 0.569i)4-s + (2.72 − 10.8i)5-s + (−1.77 + 12.3i)6-s + (−27.2 − 14.8i)7-s + (−1.70 − 7.81i)8-s + (10.8 + 4.96i)9-s + (22.0 + 3.88i)10-s + (22.6 − 19.6i)11-s + (−24.9 − 1.78i)12-s + (−1.13 − 2.08i)13-s + (25.7 − 56.4i)14-s + (30.9 − 62.5i)15-s + (15.3 − 4.50i)16-s + (79.5 − 59.5i)17-s + ⋯ |
| L(s) = 1 | + (0.0504 + 0.705i)2-s + (1.17 + 0.255i)3-s + (−0.494 + 0.0711i)4-s + (0.243 − 0.969i)5-s + (−0.120 + 0.840i)6-s + (−1.47 − 0.803i)7-s + (−0.0751 − 0.345i)8-s + (0.402 + 0.184i)9-s + (0.696 + 0.122i)10-s + (0.620 − 0.537i)11-s + (−0.599 − 0.0428i)12-s + (−0.0242 − 0.0444i)13-s + (0.492 − 1.07i)14-s + (0.533 − 1.07i)15-s + (0.239 − 0.0704i)16-s + (1.13 − 0.850i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.86245 - 0.753339i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.86245 - 0.753339i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.142 - 1.99i)T \) |
| 5 | \( 1 + (-2.72 + 10.8i)T \) |
| 23 | \( 1 + (-29.4 - 106. i)T \) |
| good | 3 | \( 1 + (-6.09 - 1.32i)T + (24.5 + 11.2i)T^{2} \) |
| 7 | \( 1 + (27.2 + 14.8i)T + (185. + 288. i)T^{2} \) |
| 11 | \( 1 + (-22.6 + 19.6i)T + (189. - 1.31e3i)T^{2} \) |
| 13 | \( 1 + (1.13 + 2.08i)T + (-1.18e3 + 1.84e3i)T^{2} \) |
| 17 | \( 1 + (-79.5 + 59.5i)T + (1.38e3 - 4.71e3i)T^{2} \) |
| 19 | \( 1 + (11.7 + 81.4i)T + (-6.58e3 + 1.93e3i)T^{2} \) |
| 29 | \( 1 + (-201. - 28.9i)T + (2.34e4 + 6.87e3i)T^{2} \) |
| 31 | \( 1 + (155. + 100. i)T + (1.23e4 + 2.70e4i)T^{2} \) |
| 37 | \( 1 + (227. - 84.7i)T + (3.82e4 - 3.31e4i)T^{2} \) |
| 41 | \( 1 + (-75.4 - 165. i)T + (-4.51e4 + 5.20e4i)T^{2} \) |
| 43 | \( 1 + (-53.1 + 244. i)T + (-7.23e4 - 3.30e4i)T^{2} \) |
| 47 | \( 1 + (-77.5 + 77.5i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + (-134. + 245. i)T + (-8.04e4 - 1.25e5i)T^{2} \) |
| 59 | \( 1 + (136. - 466. i)T + (-1.72e5 - 1.11e5i)T^{2} \) |
| 61 | \( 1 + (180. - 281. i)T + (-9.42e4 - 2.06e5i)T^{2} \) |
| 67 | \( 1 + (-673. + 48.1i)T + (2.97e5 - 4.28e4i)T^{2} \) |
| 71 | \( 1 + (-316. + 365. i)T + (-5.09e4 - 3.54e5i)T^{2} \) |
| 73 | \( 1 + (482. - 644. i)T + (-1.09e5 - 3.73e5i)T^{2} \) |
| 79 | \( 1 + (-651. - 191. i)T + (4.14e5 + 2.66e5i)T^{2} \) |
| 83 | \( 1 + (53.4 + 143. i)T + (-4.32e5 + 3.74e5i)T^{2} \) |
| 89 | \( 1 + (-848. + 545. i)T + (2.92e5 - 6.41e5i)T^{2} \) |
| 97 | \( 1 + (24.1 - 64.7i)T + (-6.89e5 - 5.97e5i)T^{2} \) |
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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
−11.90609153760149773720948454169, −10.12769215663287409204965111846, −9.346072753251239245750189528074, −8.880639104035181172236204049781, −7.70943100968699664264937361960, −6.68522713099229923618638147507, −5.42606021759933160373284407182, −3.97541607155712549954542625762, −3.09572428073085233272602028833, −0.70636868783625435540633030815,
1.96960081328551112378112098617, 3.00823052473651134108957750773, 3.68392257521987989358984593067, 5.84198263851178456772054202596, 6.82408779733472470631905600094, 8.138920267104954882435116657595, 9.165598523972705869636247663257, 9.860547526985688348718343280675, 10.68400915317113889868371102387, 12.32212997288067848692381483572
