| L(s) = 1 | + (−0.599 − 0.800i)2-s + (−1.05 + 2.81i)3-s + (−0.281 + 0.959i)4-s + (1.82 + 1.28i)5-s + (2.88 − 0.846i)6-s + (0.193 + 0.890i)7-s + (0.936 − 0.349i)8-s + (−4.55 − 3.94i)9-s + (−0.0675 − 2.23i)10-s + (−4.85 + 0.697i)11-s + (−2.40 − 1.80i)12-s + (0.517 + 0.112i)13-s + (0.597 − 0.689i)14-s + (−5.54 + 3.80i)15-s + (−0.841 − 0.540i)16-s + (5.49 + 2.99i)17-s + ⋯ |
| L(s) = 1 | + (−0.423 − 0.566i)2-s + (−0.606 + 1.62i)3-s + (−0.140 + 0.479i)4-s + (0.818 + 0.574i)5-s + (1.17 − 0.345i)6-s + (0.0732 + 0.336i)7-s + (0.331 − 0.123i)8-s + (−1.51 − 1.31i)9-s + (−0.0213 − 0.706i)10-s + (−1.46 + 0.210i)11-s + (−0.694 − 0.519i)12-s + (0.143 + 0.0312i)13-s + (0.159 − 0.184i)14-s + (−1.43 + 0.981i)15-s + (−0.210 − 0.135i)16-s + (1.33 + 0.727i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.376644 + 0.654181i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.376644 + 0.654181i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.599 + 0.800i)T \) |
| 5 | \( 1 + (-1.82 - 1.28i)T \) |
| 23 | \( 1 + (3.86 - 2.84i)T \) |
| good | 3 | \( 1 + (1.05 - 2.81i)T + (-2.26 - 1.96i)T^{2} \) |
| 7 | \( 1 + (-0.193 - 0.890i)T + (-6.36 + 2.90i)T^{2} \) |
| 11 | \( 1 + (4.85 - 0.697i)T + (10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.517 - 0.112i)T + (11.8 + 5.40i)T^{2} \) |
| 17 | \( 1 + (-5.49 - 2.99i)T + (9.19 + 14.3i)T^{2} \) |
| 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 | \( 1 + (5.80 - 0.415i)T + (36.6 - 5.26i)T^{2} \) |
| 41 | \( 1 + (-0.940 - 1.08i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-11.1 - 4.16i)T + (32.4 + 28.1i)T^{2} \) |
| 47 | \( 1 + (0.416 - 0.416i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.49 - 0.324i)T + (48.2 - 22.0i)T^{2} \) |
| 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 | \( 1 + (-3.09 + 2.31i)T + (18.8 - 64.2i)T^{2} \) |
| 71 | \( 1 + (-0.108 + 0.753i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-0.0284 - 0.0520i)T + (-39.4 + 61.4i)T^{2} \) |
| 79 | \( 1 + (-13.7 + 8.80i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-0.417 - 5.83i)T + (-82.1 + 11.8i)T^{2} \) |
| 89 | \( 1 + (2.73 + 5.99i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-0.457 + 6.39i)T + (-96.0 - 13.8i)T^{2} \) |
| 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
−12.21720843350116042339639958407, −11.14588616405672905101602835663, −10.34816473096590241951857367758, −10.06759324392780436870905599429, −9.084271545251580111034879669521, −7.83541208853085980443247350473, −5.99207879885180492517454266746, −5.25664349204381576612923705689, −3.87140784086308386173420863250, −2.56611974061885237098557427586,
0.76432549532493742252553825489, 2.27045499022386629325595052620, 5.16775816183002591433604910580, 5.83139161356168820429898565440, 6.86034643323087472834170115667, 7.83919203684611460289030201944, 8.495341087562865188618817104941, 10.02345572047416196685736505071, 10.82358333824318313069720595050, 12.26848640938192367358785873808
