| L(s) = 1 | + 0.614·2-s − 3-s − 1.62·4-s + 1.49·5-s − 0.614·6-s − 2.22·8-s + 9-s + 0.919·10-s + 11-s + 1.62·12-s − 1.27·13-s − 1.49·15-s + 1.87·16-s + 0.614·17-s + 0.614·18-s − 2.66·19-s − 2.42·20-s + 0.614·22-s + 1.87·23-s + 2.22·24-s − 2.76·25-s − 0.784·26-s − 27-s + 8.44·29-s − 0.919·30-s + 4.62·31-s + 5.60·32-s + ⋯ |
| L(s) = 1 | + 0.434·2-s − 0.577·3-s − 0.810·4-s + 0.669·5-s − 0.251·6-s − 0.787·8-s + 0.333·9-s + 0.290·10-s + 0.301·11-s + 0.468·12-s − 0.353·13-s − 0.386·15-s + 0.468·16-s + 0.149·17-s + 0.144·18-s − 0.610·19-s − 0.542·20-s + 0.131·22-s + 0.390·23-s + 0.454·24-s − 0.552·25-s − 0.153·26-s − 0.192·27-s + 1.56·29-s − 0.167·30-s + 0.830·31-s + 0.991·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.482745756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.482745756\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 0.614T + 2T^{2} \) |
| 5 | \( 1 - 1.49T + 5T^{2} \) |
| 13 | \( 1 + 1.27T + 13T^{2} \) |
| 17 | \( 1 - 0.614T + 17T^{2} \) |
| 19 | \( 1 + 2.66T + 19T^{2} \) |
| 23 | \( 1 - 1.87T + 23T^{2} \) |
| 29 | \( 1 - 8.44T + 29T^{2} \) |
| 31 | \( 1 - 4.62T + 31T^{2} \) |
| 37 | \( 1 - 0.274T + 37T^{2} \) |
| 41 | \( 1 + 4.91T + 41T^{2} \) |
| 43 | \( 1 - 9.42T + 43T^{2} \) |
| 47 | \( 1 + 3.16T + 47T^{2} \) |
| 53 | \( 1 - 6.18T + 53T^{2} \) |
| 59 | \( 1 - 1.87T + 59T^{2} \) |
| 61 | \( 1 + 8.52T + 61T^{2} \) |
| 67 | \( 1 - 5.06T + 67T^{2} \) |
| 71 | \( 1 - 16.0T + 71T^{2} \) |
| 73 | \( 1 - 8.75T + 73T^{2} \) |
| 79 | \( 1 - 9.44T + 79T^{2} \) |
| 83 | \( 1 - 9.00T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
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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
−9.515793759993932941957846410558, −8.700956683428180874441302326056, −7.86871139345985804184508635186, −6.63086465759978913014692361722, −6.09899888173562693899259610825, −5.18009075491320450875833703295, −4.58901851110983167086135011773, −3.61188434552566750565948093637, −2.36271969240122328924269905547, −0.837558579201055756746865113488,
0.837558579201055756746865113488, 2.36271969240122328924269905547, 3.61188434552566750565948093637, 4.58901851110983167086135011773, 5.18009075491320450875833703295, 6.09899888173562693899259610825, 6.63086465759978913014692361722, 7.86871139345985804184508635186, 8.700956683428180874441302326056, 9.515793759993932941957846410558
