| L(s) = 1 | − 2.65·2-s + 5.04·4-s + 1.06·5-s − 3.36·7-s − 8.08·8-s − 2.81·10-s − 11-s − 2.15·13-s + 8.94·14-s + 11.3·16-s − 3.67·17-s − 19-s + 5.34·20-s + 2.65·22-s − 3.15·23-s − 3.87·25-s + 5.73·26-s − 16.9·28-s − 7.17·29-s + 4.65·31-s − 14.0·32-s + 9.74·34-s − 3.57·35-s + 2.27·37-s + 2.65·38-s − 8.57·40-s + 11.3·41-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 2.52·4-s + 0.474·5-s − 1.27·7-s − 2.85·8-s − 0.889·10-s − 0.301·11-s − 0.599·13-s + 2.38·14-s + 2.84·16-s − 0.890·17-s − 0.229·19-s + 1.19·20-s + 0.565·22-s − 0.657·23-s − 0.775·25-s + 1.12·26-s − 3.21·28-s − 1.33·29-s + 0.835·31-s − 2.47·32-s + 1.67·34-s − 0.603·35-s + 0.373·37-s + 0.430·38-s − 1.35·40-s + 1.77·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4280313669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4280313669\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 5 | \( 1 - 1.06T + 5T^{2} \) |
| 7 | \( 1 + 3.36T + 7T^{2} \) |
| 13 | \( 1 + 2.15T + 13T^{2} \) |
| 17 | \( 1 + 3.67T + 17T^{2} \) |
| 23 | \( 1 + 3.15T + 23T^{2} \) |
| 29 | \( 1 + 7.17T + 29T^{2} \) |
| 31 | \( 1 - 4.65T + 31T^{2} \) |
| 37 | \( 1 - 2.27T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 9.38T + 43T^{2} \) |
| 47 | \( 1 - 5.77T + 47T^{2} \) |
| 53 | \( 1 - 5.65T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 - 6.98T + 61T^{2} \) |
| 67 | \( 1 + 4.81T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 + 8.08T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 9.96T + 83T^{2} \) |
| 89 | \( 1 - 4.61T + 89T^{2} \) |
| 97 | \( 1 + 4.09T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.252304432336421522553188410121, −8.735817109893949549785562938463, −7.64550271866901835071107980018, −7.19276288212777587880262687340, −6.22099752933437087802647809676, −5.79709006615324615694050315853, −4.04555108405541021466159778088, −2.67737399258020249531100234125, −2.13378401329831132052542402961, −0.54812746548476363085740567997,
0.54812746548476363085740567997, 2.13378401329831132052542402961, 2.67737399258020249531100234125, 4.04555108405541021466159778088, 5.79709006615324615694050315853, 6.22099752933437087802647809676, 7.19276288212777587880262687340, 7.64550271866901835071107980018, 8.735817109893949549785562938463, 9.252304432336421522553188410121
