| L(s) = 1 | + (−0.832 − 1.14i)2-s + (−0.342 − 1.69i)3-s + (−0.613 + 1.90i)4-s + (1.06 + 1.20i)5-s + (−1.65 + 1.80i)6-s + (0.0200 + 0.00264i)7-s + (2.68 − 0.882i)8-s + (−2.76 + 1.16i)9-s + (0.499 − 2.21i)10-s + (0.486 + 0.986i)11-s + (3.44 + 0.391i)12-s + (−1.37 − 4.03i)13-s + (−0.0137 − 0.0251i)14-s + (1.68 − 2.21i)15-s + (−3.24 − 2.33i)16-s + (−2.58 − 2.58i)17-s + ⋯ |
| L(s) = 1 | + (−0.588 − 0.808i)2-s + (−0.197 − 0.980i)3-s + (−0.306 + 0.951i)4-s + (0.474 + 0.540i)5-s + (−0.676 + 0.736i)6-s + (0.00759 + 0.000999i)7-s + (0.950 − 0.312i)8-s + (−0.921 + 0.387i)9-s + (0.157 − 0.701i)10-s + (0.146 + 0.297i)11-s + (0.993 + 0.112i)12-s + (−0.380 − 1.11i)13-s + (−0.00366 − 0.00672i)14-s + (0.436 − 0.571i)15-s + (−0.811 − 0.584i)16-s + (−0.625 − 0.625i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0898169 - 0.756653i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0898169 - 0.756653i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.832 + 1.14i)T \) |
| 3 | \( 1 + (0.342 + 1.69i)T \) |
| good | 5 | \( 1 + (-1.06 - 1.20i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (-0.0200 - 0.00264i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (-0.486 - 0.986i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (1.37 + 4.03i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (2.58 + 2.58i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.725 + 3.64i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (0.902 + 6.85i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (8.38 - 0.549i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (-4.82 + 2.78i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.768 + 3.86i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-0.345 - 2.62i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (-11.4 + 5.63i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (-0.949 + 0.254i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (5.06 - 7.57i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (-4.48 - 5.11i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (-0.656 - 10.0i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (7.84 + 3.86i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (0.699 + 1.68i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-2.18 + 5.27i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-1.72 - 6.44i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (2.44 + 2.14i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (-5.43 + 2.24i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (-2.86 - 1.65i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.53345389771334107246759570804, −9.497456231177621947214904231636, −8.624561253922471509615168210729, −7.61552949618788655928891271734, −6.98626351233124933090948769793, −5.88948601810706196591746450877, −4.55425848640737441113883858441, −2.85123237335628523111952256027, −2.23874889405235183561161914218, −0.53328008861395393944509965179,
1.69288236732809748951351813029, 3.80471681545325876989453003608, 4.83805227799114453567723832838, 5.67949404383074030028645446363, 6.44930701981105227726249983665, 7.70913148832596717126435128969, 8.712417138264589116341241520043, 9.433781446739088320708661539510, 9.821498749042149067581958404665, 10.97471985894241435793268996301
