| L(s) = 1 | + (0.0271 + 0.188i)2-s + (0.909 + 0.415i)3-s + (1.88 − 0.553i)4-s + (−1.98 + 2.29i)5-s + (−0.0537 + 0.183i)6-s + (0.170 − 2.64i)7-s + (0.314 + 0.688i)8-s + (0.654 + 0.755i)9-s + (−0.487 − 0.313i)10-s + (3.07 + 0.442i)11-s + (1.94 + 0.279i)12-s + (−1.29 + 2.01i)13-s + (0.503 − 0.0395i)14-s + (−2.76 + 1.26i)15-s + (3.18 − 2.04i)16-s + (6.24 + 1.83i)17-s + ⋯ |
| L(s) = 1 | + (0.0192 + 0.133i)2-s + (0.525 + 0.239i)3-s + (0.942 − 0.276i)4-s + (−0.889 + 1.02i)5-s + (−0.0219 + 0.0747i)6-s + (0.0643 − 0.997i)7-s + (0.111 + 0.243i)8-s + (0.218 + 0.251i)9-s + (−0.154 − 0.0991i)10-s + (0.928 + 0.133i)11-s + (0.561 + 0.0806i)12-s + (−0.359 + 0.558i)13-s + (0.134 − 0.0105i)14-s + (−0.713 + 0.325i)15-s + (0.795 − 0.511i)16-s + (1.51 + 0.444i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.82271 + 0.635814i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.82271 + 0.635814i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.909 - 0.415i)T \) |
| 7 | \( 1 + (-0.170 + 2.64i)T \) |
| 23 | \( 1 + (0.644 - 4.75i)T \) |
| good | 2 | \( 1 + (-0.0271 - 0.188i)T + (-1.91 + 0.563i)T^{2} \) |
| 5 | \( 1 + (1.98 - 2.29i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-3.07 - 0.442i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (1.29 - 2.01i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-6.24 - 1.83i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-1.82 + 0.535i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (5.49 + 1.61i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (7.97 - 3.64i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-6.32 + 5.47i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (4.97 + 4.31i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (2.26 + 1.03i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 4.70iT - 47T^{2} \) |
| 53 | \( 1 + (4.94 + 7.69i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-1.62 + 2.53i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (0.0813 + 0.178i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-12.0 + 1.73i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.06 - 7.43i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (0.328 + 1.11i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (8.23 - 12.8i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (7.53 + 8.69i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-2.14 + 4.69i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-8.54 + 9.85i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.22790008015485680635836606682, −10.21494826749127935530879079504, −9.551454615660560664185273536474, −8.011541361593379656763519959217, −7.24963237916933060362457474565, −6.94002289326704836754576731632, −5.52535488643452375738913102741, −3.83600423608206475030310329504, −3.38510649406543822349368291278, −1.71383722440686587676297386561,
1.34314363279041737222425144883, 2.83181471529128037160083038130, 3.80590299343281419809338538757, 5.20164002716481834674717647430, 6.28282726247135169269668704057, 7.59549933230508645158533437469, 8.018959770763122304516066219878, 8.990114253663170432527117419658, 9.817530924540065644701769394241, 11.24953350975035828721834443962
