| L(s) = 1 | + 1.54·2-s − 2.36·3-s + 0.395·4-s + 0.422·5-s − 3.66·6-s + 1.36·7-s − 2.48·8-s + 2.59·9-s + 0.654·10-s − 11-s − 0.935·12-s + 0.691·13-s + 2.11·14-s − 15-s − 4.63·16-s − 6.98·17-s + 4.02·18-s − 2.82·19-s + 0.167·20-s − 3.23·21-s − 1.54·22-s − 1.09·23-s + 5.87·24-s − 4.82·25-s + 1.07·26-s + 0.952·27-s + 0.540·28-s + ⋯ |
| L(s) = 1 | + 1.09·2-s − 1.36·3-s + 0.197·4-s + 0.189·5-s − 1.49·6-s + 0.516·7-s − 0.878·8-s + 0.865·9-s + 0.206·10-s − 0.301·11-s − 0.270·12-s + 0.191·13-s + 0.564·14-s − 0.258·15-s − 1.15·16-s − 1.69·17-s + 0.947·18-s − 0.648·19-s + 0.0373·20-s − 0.705·21-s − 0.329·22-s − 0.227·23-s + 1.19·24-s − 0.964·25-s + 0.209·26-s + 0.183·27-s + 0.102·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 1.54T + 2T^{2} \) |
| 3 | \( 1 + 2.36T + 3T^{2} \) |
| 5 | \( 1 - 0.422T + 5T^{2} \) |
| 7 | \( 1 - 1.36T + 7T^{2} \) |
| 13 | \( 1 - 0.691T + 13T^{2} \) |
| 17 | \( 1 + 6.98T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 1.09T + 23T^{2} \) |
| 29 | \( 1 + 0.882T + 29T^{2} \) |
| 31 | \( 1 + 2.77T + 31T^{2} \) |
| 37 | \( 1 + 8.28T + 37T^{2} \) |
| 41 | \( 1 + 2.27T + 41T^{2} \) |
| 43 | \( 1 + 0.00667T + 43T^{2} \) |
| 47 | \( 1 - 5.49T + 47T^{2} \) |
| 53 | \( 1 - 5.40T + 53T^{2} \) |
| 59 | \( 1 - 4.46T + 59T^{2} \) |
| 67 | \( 1 + 8.41T + 67T^{2} \) |
| 71 | \( 1 - 2.07T + 71T^{2} \) |
| 73 | \( 1 - 9.89T + 73T^{2} \) |
| 79 | \( 1 - 9.51T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 - 1.19T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−10.52674840883889547610349222184, −9.261845423398278003158265631050, −8.348604642904241548479672150700, −6.90349246449542480006631895130, −6.19305739402128597098919339412, −5.39958313256199352887360687871, −4.72823837303605945614262468954, −3.84736500072943321524358003967, −2.16492764116722575854130681317, 0,
2.16492764116722575854130681317, 3.84736500072943321524358003967, 4.72823837303605945614262468954, 5.39958313256199352887360687871, 6.19305739402128597098919339412, 6.90349246449542480006631895130, 8.348604642904241548479672150700, 9.261845423398278003158265631050, 10.52674840883889547610349222184
