| L(s) = 1 | + (−0.997 + 0.0713i)2-s + (−0.212 + 0.977i)3-s + (0.989 − 0.142i)4-s + (−0.619 − 2.14i)5-s + (0.142 − 0.989i)6-s + (−1.99 + 3.65i)7-s + (−0.977 + 0.212i)8-s + (−0.909 − 0.415i)9-s + (0.771 + 2.09i)10-s + (1.81 − 1.57i)11-s + (−0.0713 + 0.997i)12-s + (−2.13 + 1.16i)13-s + (1.72 − 3.78i)14-s + (2.23 − 0.148i)15-s + (0.959 − 0.281i)16-s + (1.26 + 1.68i)17-s + ⋯ |
| L(s) = 1 | + (−0.705 + 0.0504i)2-s + (−0.122 + 0.564i)3-s + (0.494 − 0.0711i)4-s + (−0.276 − 0.960i)5-s + (0.0580 − 0.404i)6-s + (−0.754 + 1.38i)7-s + (−0.345 + 0.0751i)8-s + (−0.303 − 0.138i)9-s + (0.243 + 0.663i)10-s + (0.546 − 0.473i)11-s + (−0.0205 + 0.287i)12-s + (−0.590 + 0.322i)13-s + (0.462 − 1.01i)14-s + (0.576 − 0.0383i)15-s + (0.239 − 0.0704i)16-s + (0.306 + 0.408i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0979034 - 0.196057i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0979034 - 0.196057i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.997 - 0.0713i)T \) |
| 3 | \( 1 + (0.212 - 0.977i)T \) |
| 5 | \( 1 + (0.619 + 2.14i)T \) |
| 23 | \( 1 + (0.560 + 4.76i)T \) |
| good | 7 | \( 1 + (1.99 - 3.65i)T + (-3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-1.81 + 1.57i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (2.13 - 1.16i)T + (7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-1.26 - 1.68i)T + (-4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.707 + 4.91i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (3.75 + 0.539i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (8.69 + 5.58i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (0.758 + 2.03i)T + (-27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (-3.56 - 7.80i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.874 - 0.190i)T + (39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (5.40 + 5.40i)T + 47iT^{2} \) |
| 53 | \( 1 + (8.38 + 4.57i)T + (28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-3.47 + 11.8i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (6.51 - 10.1i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (0.779 + 10.9i)T + (-66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-3.90 + 4.50i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (5.19 + 3.88i)T + (20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-0.848 - 0.249i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (2.04 - 0.763i)T + (62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (10.1 - 6.55i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-13.2 - 4.95i)T + (73.3 + 63.5i)T^{2} \) |
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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
−9.773050993646610986047766831520, −9.215293679051748394226074234054, −8.801865123244252114215652047179, −7.83326954825524281795360872211, −6.50038821844403328701529365333, −5.72957014526042749983101611831, −4.75132204195580108643633201969, −3.45859063323898972051505994773, −2.13940319492393429685535148958, −0.14356809184549239825035597924,
1.57332457599876177124798424156, 3.12206267834804484499013296392, 3.95081695011191195232297097273, 5.72930002298725719077676081622, 6.77512690213800758656504030408, 7.34464441874215694627380473271, 7.75569423427808282689776188077, 9.242875503753526103900826756561, 10.03389698656185714125798253670, 10.60465643082780062229991959974
