L(s) = 1 | + (−1 + 1.73i)2-s + (−0.999 − 1.73i)4-s − 4·7-s + (1.5 + 2.59i)9-s − 11-s + (1 + 1.73i)13-s + (4 − 6.92i)14-s + (1.99 − 3.46i)16-s + (1 − 1.73i)17-s − 6·18-s + (−3.5 − 2.59i)19-s + (1 − 1.73i)22-s + (−3 − 5.19i)23-s − 3.99·26-s + (3.99 + 6.92i)28-s + (−4.5 − 7.79i)29-s + ⋯ |
L(s) = 1 | + (−0.707 + 1.22i)2-s + (−0.499 − 0.866i)4-s − 1.51·7-s + (0.5 + 0.866i)9-s − 0.301·11-s + (0.277 + 0.480i)13-s + (1.06 − 1.85i)14-s + (0.499 − 0.866i)16-s + (0.242 − 0.420i)17-s − 1.41·18-s + (−0.802 − 0.596i)19-s + (0.213 − 0.369i)22-s + (−0.625 − 1.08i)23-s − 0.784·26-s + (0.755 + 1.30i)28-s + (−0.835 − 1.44i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (3.5 + 2.59i)T \) |
good | 2 | \( 1 + (1 - 1.73i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (1 - 1.73i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (-5.5 - 9.52i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3 + 5.19i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.36751164086513074431839345817, −9.716328079682888654030215313233, −8.910473288127111086879216213944, −7.977194035414614945938543856948, −7.07799198823784898065570791569, −6.43332306229834107196970330317, −5.51485640114720601559545485755, −4.10228078569520378231572376488, −2.55355022563390817707266768711, 0,
1.68095685129292630338843942599, 3.29013321991267835158951725753, 3.70399542926455084123582712884, 5.75222278372115589474196927489, 6.56685710101383599199555415969, 7.81680258989507878547397959837, 9.061086989972332851215588596742, 9.532024088411233542618671296969, 10.30600777154511825610365359512