| L(s) = 1 | + 2-s − 1.18·3-s + 4-s − 5-s − 1.18·6-s + 4.68·7-s + 8-s − 1.59·9-s − 10-s + 1.20·11-s − 1.18·12-s + 5.57·13-s + 4.68·14-s + 1.18·15-s + 16-s + 5.07·17-s − 1.59·18-s + 7.66·19-s − 20-s − 5.55·21-s + 1.20·22-s − 6.33·23-s − 1.18·24-s + 25-s + 5.57·26-s + 5.44·27-s + 4.68·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.683·3-s + 0.5·4-s − 0.447·5-s − 0.483·6-s + 1.77·7-s + 0.353·8-s − 0.532·9-s − 0.316·10-s + 0.363·11-s − 0.341·12-s + 1.54·13-s + 1.25·14-s + 0.305·15-s + 0.250·16-s + 1.22·17-s − 0.376·18-s + 1.75·19-s − 0.223·20-s − 1.21·21-s + 0.256·22-s − 1.32·23-s − 0.241·24-s + 0.200·25-s + 1.09·26-s + 1.04·27-s + 0.885·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.083557714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.083557714\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 + 1.18T + 3T^{2} \) |
| 7 | \( 1 - 4.68T + 7T^{2} \) |
| 11 | \( 1 - 1.20T + 11T^{2} \) |
| 13 | \( 1 - 5.57T + 13T^{2} \) |
| 17 | \( 1 - 5.07T + 17T^{2} \) |
| 19 | \( 1 - 7.66T + 19T^{2} \) |
| 23 | \( 1 + 6.33T + 23T^{2} \) |
| 29 | \( 1 + 0.292T + 29T^{2} \) |
| 31 | \( 1 + 8.91T + 31T^{2} \) |
| 37 | \( 1 + 4.02T + 37T^{2} \) |
| 41 | \( 1 + 4.54T + 41T^{2} \) |
| 43 | \( 1 - 3.79T + 43T^{2} \) |
| 47 | \( 1 + 6.89T + 47T^{2} \) |
| 53 | \( 1 + 2.85T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 8.87T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 5.98T + 71T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 - 1.14T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−8.219110929700780764027884246631, −7.75224669957796466323779420272, −6.94709800142517173388449293318, −5.79656116109606565812776675429, −5.55374426256635729097445726648, −4.84535831234805464773309146821, −3.84903649849920701254375007977, −3.30467331546262947556021740821, −1.79765876892825370598907773308, −1.03863944916582558556592419302,
1.03863944916582558556592419302, 1.79765876892825370598907773308, 3.30467331546262947556021740821, 3.84903649849920701254375007977, 4.84535831234805464773309146821, 5.55374426256635729097445726648, 5.79656116109606565812776675429, 6.94709800142517173388449293318, 7.75224669957796466323779420272, 8.219110929700780764027884246631
