| L(s) = 1 | + (0.989 + 0.142i)2-s + (−0.909 + 0.415i)3-s + (0.959 + 0.281i)4-s + (−2.14 + 0.635i)5-s + (−0.959 + 0.281i)6-s + (2.47 − 3.84i)7-s + (0.909 + 0.415i)8-s + (0.654 − 0.755i)9-s + (−2.21 + 0.324i)10-s + (−0.00211 − 0.0146i)11-s + (−0.989 + 0.142i)12-s + (−1.93 − 3.00i)13-s + (2.99 − 3.45i)14-s + (1.68 − 1.46i)15-s + (0.841 + 0.540i)16-s + (−0.817 − 2.78i)17-s + ⋯ |
| L(s) = 1 | + (0.699 + 0.100i)2-s + (−0.525 + 0.239i)3-s + (0.479 + 0.140i)4-s + (−0.958 + 0.284i)5-s + (−0.391 + 0.115i)6-s + (0.935 − 1.45i)7-s + (0.321 + 0.146i)8-s + (0.218 − 0.251i)9-s + (−0.699 + 0.102i)10-s + (−0.000636 − 0.00442i)11-s + (−0.285 + 0.0410i)12-s + (−0.535 − 0.833i)13-s + (0.800 − 0.924i)14-s + (0.435 − 0.379i)15-s + (0.210 + 0.135i)16-s + (−0.198 − 0.675i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 + 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.62105 - 0.598434i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.62105 - 0.598434i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.989 - 0.142i)T \) |
| 3 | \( 1 + (0.909 - 0.415i)T \) |
| 5 | \( 1 + (2.14 - 0.635i)T \) |
| 23 | \( 1 + (4.29 + 2.13i)T \) |
| good | 7 | \( 1 + (-2.47 + 3.84i)T + (-2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.00211 + 0.0146i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (1.93 + 3.00i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (0.817 + 2.78i)T + (-14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-7.13 - 2.09i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-6.47 + 1.90i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-1.24 + 2.73i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-2.52 - 2.18i)T + (5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (2.38 + 2.74i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-6.63 + 3.03i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 0.920iT - 47T^{2} \) |
| 53 | \( 1 + (4.07 - 6.33i)T + (-22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-7.23 + 4.65i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (2.74 - 6.00i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (5.29 + 0.761i)T + (64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (1.86 - 12.9i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-1.35 + 4.60i)T + (-61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (13.4 - 8.61i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (10.6 + 9.20i)T + (11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (3.94 + 8.63i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (4.85 - 4.20i)T + (13.8 - 96.0i)T^{2} \) |
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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
−10.48849324572319055420189023229, −9.972835269966413493065998496475, −8.226408955419070021579877357696, −7.53718919609539071171856054145, −7.02270344507673500936700273800, −5.68881706160744532142463784842, −4.65505966623463233008130563172, −4.12340027355854188820349222862, −2.97195754575384451098378997849, −0.855558543519269342941637418599,
1.58511846444080476721413516991, 2.90484662957998336993838503831, 4.36202142109607618028921127398, 5.04398611909060675256129360696, 5.85441801735119326793970152020, 6.98798662244296271192248292223, 7.892169868583885201105378259969, 8.687244836320934015560988621031, 9.735962341304586060623986855189, 11.09907946748460835799624198256
