| L(s) = 1 | + (0.800 − 0.599i)2-s + (2.81 + 1.05i)3-s + (0.281 − 0.959i)4-s + (−1.01 − 1.99i)5-s + (2.88 − 0.846i)6-s + (−0.890 + 0.193i)7-s + (−0.349 − 0.936i)8-s + (4.55 + 3.94i)9-s + (−2.00 − 0.989i)10-s + (−4.85 + 0.697i)11-s + (1.80 − 2.40i)12-s + (0.112 − 0.517i)13-s + (−0.597 + 0.689i)14-s + (−0.754 − 6.67i)15-s + (−0.841 − 0.540i)16-s + (−2.99 + 5.49i)17-s + ⋯ |
| L(s) = 1 | + (0.566 − 0.423i)2-s + (1.62 + 0.606i)3-s + (0.140 − 0.479i)4-s + (−0.452 − 0.891i)5-s + (1.17 − 0.345i)6-s + (−0.336 + 0.0732i)7-s + (−0.123 − 0.331i)8-s + (1.51 + 1.31i)9-s + (−0.634 − 0.313i)10-s + (−1.46 + 0.210i)11-s + (0.519 − 0.694i)12-s + (0.0312 − 0.143i)13-s + (−0.159 + 0.184i)14-s + (−0.194 − 1.72i)15-s + (−0.210 − 0.135i)16-s + (−0.727 + 1.33i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.21540 - 0.479729i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.21540 - 0.479729i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.800 + 0.599i)T \) |
| 5 | \( 1 + (1.01 + 1.99i)T \) |
| 23 | \( 1 + (-2.84 - 3.86i)T \) |
| good | 3 | \( 1 + (-2.81 - 1.05i)T + (2.26 + 1.96i)T^{2} \) |
| 7 | \( 1 + (0.890 - 0.193i)T + (6.36 - 2.90i)T^{2} \) |
| 11 | \( 1 + (4.85 - 0.697i)T + (10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.112 + 0.517i)T + (-11.8 - 5.40i)T^{2} \) |
| 17 | \( 1 + (2.99 - 5.49i)T + (-9.19 - 14.3i)T^{2} \) |
| 19 | \( 1 + (-3.07 - 0.901i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (0.421 + 1.43i)T + (-24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.23 + 7.07i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (0.415 + 5.80i)T + (-36.6 + 5.26i)T^{2} \) |
| 41 | \( 1 + (-0.940 - 1.08i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-4.16 + 11.1i)T + (-32.4 - 28.1i)T^{2} \) |
| 47 | \( 1 + (0.416 + 0.416i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.324 - 1.49i)T + (-48.2 + 22.0i)T^{2} \) |
| 59 | \( 1 + (7.41 + 11.5i)T + (-24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-5.26 - 2.40i)T + (39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-2.31 - 3.09i)T + (-18.8 + 64.2i)T^{2} \) |
| 71 | \( 1 + (-0.108 + 0.753i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-0.0520 + 0.0284i)T + (39.4 - 61.4i)T^{2} \) |
| 79 | \( 1 + (13.7 - 8.80i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-5.83 + 0.417i)T + (82.1 - 11.8i)T^{2} \) |
| 89 | \( 1 + (-2.73 - 5.99i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-6.39 - 0.457i)T + (96.0 + 13.8i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.57678609851820231384190106765, −11.09131862130627867388626769145, −10.05799353114214232528196919630, −9.268897416833266252110275404941, −8.294184861556344121680787329792, −7.54683973844211234981196291390, −5.51213404072315407360297377893, −4.35986497152675152618960852645, −3.45267422350098613744982258033, −2.19513657742356317100546671656,
2.74074648020749611654462935948, 3.08936713396127114798468041350, 4.72991559468434336405173808152, 6.59202018094411160036381359557, 7.32293553301139804599388029481, 8.039936043358914893334778042347, 9.039886844741688768244125685413, 10.26300896346339750840494443166, 11.48782596789695233206615896221, 12.71598251177029289032742494241
