L(s) = 1 | + (1.03 + 1.43i)2-s + (−0.348 + 1.07i)4-s + (−2.51 − 0.817i)7-s + (1.46 − 0.476i)8-s + (−2.42 + 1.76i)9-s + (−0.0629 − 3.31i)11-s + (−1.44 − 4.45i)14-s + (4.03 + 2.92i)16-s + (−5.04 − 1.63i)18-s + (4.67 − 3.53i)22-s + 2.56·23-s + (1.54 + 4.75i)25-s + (1.75 − 2.41i)28-s + (−7.02 − 2.28i)29-s + 5.72i·32-s + ⋯ |
L(s) = 1 | + (0.735 + 1.01i)2-s + (−0.174 + 0.536i)4-s + (−0.951 − 0.309i)7-s + (0.518 − 0.168i)8-s + (−0.809 + 0.587i)9-s + (−0.0189 − 0.999i)11-s + (−0.386 − 1.18i)14-s + (1.00 + 0.732i)16-s + (−1.18 − 0.386i)18-s + (0.997 − 0.754i)22-s + 0.533·23-s + (0.309 + 0.951i)25-s + (0.331 − 0.456i)28-s + (−1.30 − 0.423i)29-s + 1.01i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.513 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07590 + 0.609592i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07590 + 0.609592i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2.51 + 0.817i)T \) |
| 11 | \( 1 + (0.0629 + 3.31i)T \) |
good | 2 | \( 1 + (-1.03 - 1.43i)T + (-0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (2.42 - 1.76i)T^{2} \) |
| 5 | \( 1 + (-1.54 - 4.75i)T^{2} \) |
| 13 | \( 1 + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 2.56T + 23T^{2} \) |
| 29 | \( 1 + (7.02 + 2.28i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (3.68 - 11.3i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 2.77iT - 43T^{2} \) |
| 47 | \( 1 + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-11.5 + 8.41i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 + (12.9 + 9.43i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (10.3 + 14.1i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (-29.9 + 92.2i)T^{2} \) |
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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
−14.66938901233728871005831468028, −13.58064875398064227331672669574, −13.14982682131617767900249354791, −11.43537956939211309199009897467, −10.28723014138534710875160056948, −8.730210530391355368258132373125, −7.39903797844460663265499732152, −6.22382220924576802860251920454, −5.23099003626911556878673020908, −3.44143414793870589705821960940,
2.58569582382856627395194458269, 3.92126450689799338293445164155, 5.57137144896879636119229479647, 7.14377303655170019470895158886, 8.941444054078672315956105000880, 10.09028318527243042282111084493, 11.27661089555602973467197247061, 12.35487537936776652893335292887, 12.83496210404122822679605629400, 14.10952473240440264594916999953