| L(s) = 1 | + (1.06 − 0.931i)2-s + (1.31 − 1.13i)3-s + (0.264 − 1.98i)4-s + (2.27 + 2.59i)5-s + (0.342 − 2.42i)6-s + (−1.60 − 0.211i)7-s + (−1.56 − 2.35i)8-s + (0.441 − 2.96i)9-s + (4.83 + 0.640i)10-s + (1.80 + 3.65i)11-s + (−1.89 − 2.89i)12-s + (−0.973 − 2.86i)13-s + (−1.90 + 1.27i)14-s + (5.91 + 0.828i)15-s + (−3.86 − 1.04i)16-s + (1.07 + 1.07i)17-s + ⋯ |
| L(s) = 1 | + (0.752 − 0.658i)2-s + (0.757 − 0.653i)3-s + (0.132 − 0.991i)4-s + (1.01 + 1.15i)5-s + (0.139 − 0.990i)6-s + (−0.606 − 0.0798i)7-s + (−0.553 − 0.832i)8-s + (0.147 − 0.989i)9-s + (1.52 + 0.202i)10-s + (0.543 + 1.10i)11-s + (−0.547 − 0.837i)12-s + (−0.270 − 0.795i)13-s + (−0.508 + 0.339i)14-s + (1.52 + 0.214i)15-s + (−0.965 − 0.262i)16-s + (0.261 + 0.261i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.227 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.36306 - 1.87551i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.36306 - 1.87551i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.06 + 0.931i)T \) |
| 3 | \( 1 + (-1.31 + 1.13i)T \) |
| good | 5 | \( 1 + (-2.27 - 2.59i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (1.60 + 0.211i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (-1.80 - 3.65i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (0.973 + 2.86i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (-1.07 - 1.07i)T + 17iT^{2} \) |
| 19 | \( 1 + (-1.23 + 6.19i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-0.0343 - 0.260i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (-0.590 + 0.0387i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (6.42 - 3.70i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.25 - 11.3i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-1.59 - 12.1i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (-5.57 + 2.74i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (2.98 - 0.799i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (5.17 - 7.73i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (5.38 + 6.14i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (0.313 + 4.78i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (1.06 + 0.522i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (4.32 + 10.4i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (0.785 - 1.89i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-1.57 - 5.88i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (2.84 + 2.49i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (-16.7 + 6.95i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (6.95 + 4.01i)T + (48.5 + 84.0i)T^{2} \) |
| 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
−10.46761057812311428815332162644, −9.709967452719626847885536789105, −9.273660695218721189473571173494, −7.55725539997057008398989892321, −6.66250856165533007043046061776, −6.23023333044319221720770916225, −4.81076815057699261477706384062, −3.26804242967800643050460443718, −2.74234585340186693228516151131, −1.59121110883276186673658974536,
2.05531450906780580706547721358, 3.47723884640923923288855372585, 4.28890407241168593739325267434, 5.52119697496680694393466647587, 5.94101878829380832039178229609, 7.35498601409870745261633510201, 8.409578683995438041989646857987, 9.179578179963621992287787311785, 9.545302418804459905225097823551, 10.88646691502091790497914947297
