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
−10.60465643082780062229991959974, −10.03389698656185714125798253670, −9.242875503753526103900826756561, −7.75569423427808282689776188077, −7.34464441874215694627380473271, −6.77512690213800758656504030408, −5.72930002298725719077676081622, −3.95081695011191195232297097273, −3.12206267834804484499013296392, −1.57332457599876177124798424156,
0.14356809184549239825035597924, 2.13940319492393429685535148958, 3.45859063323898972051505994773, 4.75132204195580108643633201969, 5.72957014526042749983101611831, 6.50038821844403328701529365333, 7.83326954825524281795360872211, 8.801865123244252114215652047179, 9.215293679051748394226074234054, 9.773050993646610986047766831520