| L(s) = 1 | + (−1.37 + 1.04i)3-s + (−1.84 − 1.26i)5-s + (2.63 + 0.269i)7-s + (0.801 − 2.89i)9-s + 3.01i·11-s − 4.39·13-s + (3.86 − 0.180i)15-s + 2.71i·17-s − 8.23i·19-s + (−3.91 + 2.38i)21-s + 3.81·23-s + (1.77 + 4.67i)25-s + (1.92 + 4.82i)27-s + 1.17i·29-s + 1.73i·31-s + ⋯ |
| L(s) = 1 | + (−0.796 + 0.605i)3-s + (−0.823 − 0.567i)5-s + (0.994 + 0.101i)7-s + (0.267 − 0.963i)9-s + 0.908i·11-s − 1.21·13-s + (0.998 − 0.0465i)15-s + 0.657i·17-s − 1.88i·19-s + (−0.853 + 0.521i)21-s + 0.796·23-s + (0.355 + 0.934i)25-s + (0.370 + 0.928i)27-s + 0.217i·29-s + 0.311i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1964824507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1964824507\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.37 - 1.04i)T \) |
| 5 | \( 1 + (1.84 + 1.26i)T \) |
| 7 | \( 1 + (-2.63 - 0.269i)T \) |
| good | 11 | \( 1 - 3.01iT - 11T^{2} \) |
| 13 | \( 1 + 4.39T + 13T^{2} \) |
| 17 | \( 1 - 2.71iT - 17T^{2} \) |
| 19 | \( 1 + 8.23iT - 19T^{2} \) |
| 23 | \( 1 - 3.81T + 23T^{2} \) |
| 29 | \( 1 - 1.17iT - 29T^{2} \) |
| 31 | \( 1 - 1.73iT - 31T^{2} \) |
| 37 | \( 1 + 4.60iT - 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 9.18iT - 43T^{2} \) |
| 47 | \( 1 - 12.0iT - 47T^{2} \) |
| 53 | \( 1 + 7.14T + 53T^{2} \) |
| 59 | \( 1 + 9.11T + 59T^{2} \) |
| 61 | \( 1 + 13.5iT - 61T^{2} \) |
| 67 | \( 1 - 0.494iT - 67T^{2} \) |
| 71 | \( 1 + 5.15iT - 71T^{2} \) |
| 73 | \( 1 + 4.76T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 8.16iT - 83T^{2} \) |
| 89 | \( 1 + 2.26T + 89T^{2} \) |
| 97 | \( 1 + 2.92T + 97T^{2} \) |
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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
−9.576567049903134696101726428966, −9.148652689917877881616906706191, −8.128964598144394696511689490270, −7.33334021527637264897717657935, −6.64792799611507070375589219828, −5.24683020410519126640417075406, −4.76457619672106370828916224305, −4.36245669281851698187626257715, −2.96113653728955617608613863341, −1.41959101673510190813816344021,
0.087892658315008877618835853126, 1.53368683644994424838630343374, 2.79095787794729195038636392473, 3.98068440528339042758922783758, 4.98589286275465228357324336225, 5.62952604041359848867549563597, 6.72293105985633203939046184022, 7.37666303343448760386088073992, 7.981118526525972296147817796587, 8.646280802443770993362811585561
