| L(s) = 1 | + 2.33·2-s − 2.90·3-s + 3.45·4-s + 1.62·5-s − 6.77·6-s + 3.07·7-s + 3.39·8-s + 5.42·9-s + 3.79·10-s + 4.58·11-s − 10.0·12-s + 3.94·13-s + 7.17·14-s − 4.71·15-s + 1.02·16-s − 2.33·17-s + 12.6·18-s + 4.12·19-s + 5.60·20-s − 8.91·21-s + 10.7·22-s − 4.52·23-s − 9.85·24-s − 2.36·25-s + 9.21·26-s − 7.02·27-s + 10.6·28-s + ⋯ |
| L(s) = 1 | + 1.65·2-s − 1.67·3-s + 1.72·4-s + 0.726·5-s − 2.76·6-s + 1.16·7-s + 1.20·8-s + 1.80·9-s + 1.19·10-s + 1.38·11-s − 2.89·12-s + 1.09·13-s + 1.91·14-s − 1.21·15-s + 0.255·16-s − 0.566·17-s + 2.98·18-s + 0.946·19-s + 1.25·20-s − 1.94·21-s + 2.28·22-s − 0.943·23-s − 2.01·24-s − 0.472·25-s + 1.80·26-s − 1.35·27-s + 2.00·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.111200125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.111200125\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2671 | \( 1 - T \) |
| good | 2 | \( 1 - 2.33T + 2T^{2} \) |
| 3 | \( 1 + 2.90T + 3T^{2} \) |
| 5 | \( 1 - 1.62T + 5T^{2} \) |
| 7 | \( 1 - 3.07T + 7T^{2} \) |
| 11 | \( 1 - 4.58T + 11T^{2} \) |
| 13 | \( 1 - 3.94T + 13T^{2} \) |
| 17 | \( 1 + 2.33T + 17T^{2} \) |
| 19 | \( 1 - 4.12T + 19T^{2} \) |
| 23 | \( 1 + 4.52T + 23T^{2} \) |
| 29 | \( 1 + 2.90T + 29T^{2} \) |
| 31 | \( 1 + 1.57T + 31T^{2} \) |
| 37 | \( 1 - 8.02T + 37T^{2} \) |
| 41 | \( 1 + 0.390T + 41T^{2} \) |
| 43 | \( 1 + 1.93T + 43T^{2} \) |
| 47 | \( 1 - 3.68T + 47T^{2} \) |
| 53 | \( 1 - 1.97T + 53T^{2} \) |
| 59 | \( 1 + 6.45T + 59T^{2} \) |
| 61 | \( 1 - 3.50T + 61T^{2} \) |
| 67 | \( 1 + 8.45T + 67T^{2} \) |
| 71 | \( 1 - 8.09T + 71T^{2} \) |
| 73 | \( 1 - 6.20T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 1.54T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−8.946070721020781198659781130170, −7.67467500207152978270719178077, −6.76070611827736172252914320920, −6.04669785563192007956835274918, −5.84010224640267388505541638615, −4.99815188550644785435238178273, −4.33308662720029054419742374282, −3.67548723940313382719797409821, −2.01212069961549519843666687726, −1.23214420751125013671872249933,
1.23214420751125013671872249933, 2.01212069961549519843666687726, 3.67548723940313382719797409821, 4.33308662720029054419742374282, 4.99815188550644785435238178273, 5.84010224640267388505541638615, 6.04669785563192007956835274918, 6.76070611827736172252914320920, 7.67467500207152978270719178077, 8.946070721020781198659781130170
