| L(s) = 1 | + 0.185·2-s − 3-s − 1.96·4-s − 5-s − 0.185·6-s + 4.18·7-s − 0.734·8-s + 9-s − 0.185·10-s + 2.22·11-s + 1.96·12-s + 2.71·13-s + 0.775·14-s + 15-s + 3.79·16-s − 4.35·17-s + 0.185·18-s + 6.68·19-s + 1.96·20-s − 4.18·21-s + 0.411·22-s + 0.734·24-s + 25-s + 0.501·26-s − 27-s − 8.22·28-s + 5.07·29-s + ⋯ |
| L(s) = 1 | + 0.130·2-s − 0.577·3-s − 0.982·4-s − 0.447·5-s − 0.0756·6-s + 1.58·7-s − 0.259·8-s + 0.333·9-s − 0.0585·10-s + 0.669·11-s + 0.567·12-s + 0.751·13-s + 0.207·14-s + 0.258·15-s + 0.948·16-s − 1.05·17-s + 0.0436·18-s + 1.53·19-s + 0.439·20-s − 0.913·21-s + 0.0877·22-s + 0.149·24-s + 0.200·25-s + 0.0984·26-s − 0.192·27-s − 1.55·28-s + 0.943·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.880321510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.880321510\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 0.185T + 2T^{2} \) |
| 7 | \( 1 - 4.18T + 7T^{2} \) |
| 11 | \( 1 - 2.22T + 11T^{2} \) |
| 13 | \( 1 - 2.71T + 13T^{2} \) |
| 17 | \( 1 + 4.35T + 17T^{2} \) |
| 19 | \( 1 - 6.68T + 19T^{2} \) |
| 29 | \( 1 - 5.07T + 29T^{2} \) |
| 31 | \( 1 - 2.45T + 31T^{2} \) |
| 37 | \( 1 - 5.28T + 37T^{2} \) |
| 41 | \( 1 + 5.40T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 2.26T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 - 6.09T + 59T^{2} \) |
| 61 | \( 1 - 4.75T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 5.07T + 73T^{2} \) |
| 79 | \( 1 + 9.20T + 79T^{2} \) |
| 83 | \( 1 + 8.87T + 83T^{2} \) |
| 89 | \( 1 - 0.783T + 89T^{2} \) |
| 97 | \( 1 + 1.06T + 97T^{2} \) |
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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
−7.899102296349078216987996929399, −7.23309401625254886679956709718, −6.38527176747768382542124397570, −5.49326998856904642105292646806, −5.06390483779978186809478629224, −4.19057590084962361615373978136, −4.02744758914941745836332062333, −2.73902449962939508582679051889, −1.38306748599214929701602362658, −0.810183383774249264420969817714,
0.810183383774249264420969817714, 1.38306748599214929701602362658, 2.73902449962939508582679051889, 4.02744758914941745836332062333, 4.19057590084962361615373978136, 5.06390483779978186809478629224, 5.49326998856904642105292646806, 6.38527176747768382542124397570, 7.23309401625254886679956709718, 7.899102296349078216987996929399
