| L(s) = 1 | + (−0.142 + 0.989i)2-s + (−0.415 − 0.909i)3-s + (−0.959 − 0.281i)4-s + (0.654 + 0.755i)5-s + (0.959 − 0.281i)6-s + (−1.25 − 0.807i)7-s + (0.415 − 0.909i)8-s + (−0.654 + 0.755i)9-s + (−0.841 + 0.540i)10-s + (−0.517 − 3.59i)11-s + (0.142 + 0.989i)12-s + (−2.43 + 1.56i)13-s + (0.978 − 1.12i)14-s + (0.415 − 0.909i)15-s + (0.841 + 0.540i)16-s + (−2.69 + 0.790i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (−0.239 − 0.525i)3-s + (−0.479 − 0.140i)4-s + (0.292 + 0.337i)5-s + (0.391 − 0.115i)6-s + (−0.475 − 0.305i)7-s + (0.146 − 0.321i)8-s + (−0.218 + 0.251i)9-s + (−0.266 + 0.170i)10-s + (−0.156 − 1.08i)11-s + (0.0410 + 0.285i)12-s + (−0.674 + 0.433i)13-s + (0.261 − 0.301i)14-s + (0.107 − 0.234i)15-s + (0.210 + 0.135i)16-s + (−0.653 + 0.191i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.487 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.202557 - 0.345210i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.202557 - 0.345210i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 3 | \( 1 + (0.415 + 0.909i)T \) |
| 5 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (3.70 + 3.04i)T \) |
| good | 7 | \( 1 + (1.25 + 0.807i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (0.517 + 3.59i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (2.43 - 1.56i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (2.69 - 0.790i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (1.45 + 0.426i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (5.62 - 1.65i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.97 + 6.52i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-0.584 + 0.675i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (2.84 + 3.28i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-0.580 - 1.27i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 5.18T + 47T^{2} \) |
| 53 | \( 1 + (6.33 + 4.07i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (2.60 - 1.67i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.11 + 6.82i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (0.328 - 2.28i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (0.0472 - 0.328i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (6.08 + 1.78i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-1.62 + 1.04i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-7.84 + 9.05i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-2.81 - 6.16i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-9.31 - 10.7i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.11813913524603557122053823994, −9.228086597819549329144747706325, −8.298246648240880729390734709702, −7.45664449973573588832673938641, −6.50021016556454405722465812069, −6.07394428594946044025389352818, −4.89539860143924617624133708041, −3.62887370767223210002561502442, −2.18298156376843837445690201229, −0.21082511393962843399630122310,
1.89008857898561734658137260739, 3.06451021222530027226273541382, 4.34590529032605507768322185275, 5.08102988185201309141271177192, 6.13077668712216917401550516732, 7.32961833630210711873604104781, 8.418432162341023867208963702157, 9.450224939898825854593928952842, 9.813314338003080218422054705787, 10.60541538644977991319500704009
