| L(s) = 1 | + 2-s + 4-s − 0.166·5-s − 2.36·7-s + 8-s − 0.166·10-s − 6.37·11-s + 4.95·13-s − 2.36·14-s + 16-s + 3.42·17-s − 0.166·20-s − 6.37·22-s + 2.68·23-s − 4.97·25-s + 4.95·26-s − 2.36·28-s + 0.736·29-s + 0.258·31-s + 32-s + 3.42·34-s + 0.394·35-s − 1.42·37-s − 0.166·40-s + 4.91·41-s − 8.23·43-s − 6.37·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.0746·5-s − 0.892·7-s + 0.353·8-s − 0.0528·10-s − 1.92·11-s + 1.37·13-s − 0.630·14-s + 0.250·16-s + 0.830·17-s − 0.0373·20-s − 1.35·22-s + 0.560·23-s − 0.994·25-s + 0.971·26-s − 0.446·28-s + 0.136·29-s + 0.0464·31-s + 0.176·32-s + 0.587·34-s + 0.0666·35-s − 0.234·37-s − 0.0264·40-s + 0.766·41-s − 1.25·43-s − 0.961·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.507526067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.507526067\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 0.166T + 5T^{2} \) |
| 7 | \( 1 + 2.36T + 7T^{2} \) |
| 11 | \( 1 + 6.37T + 11T^{2} \) |
| 13 | \( 1 - 4.95T + 13T^{2} \) |
| 17 | \( 1 - 3.42T + 17T^{2} \) |
| 23 | \( 1 - 2.68T + 23T^{2} \) |
| 29 | \( 1 - 0.736T + 29T^{2} \) |
| 31 | \( 1 - 0.258T + 31T^{2} \) |
| 37 | \( 1 + 1.42T + 37T^{2} \) |
| 41 | \( 1 - 4.91T + 41T^{2} \) |
| 43 | \( 1 + 8.23T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 + 3.17T + 59T^{2} \) |
| 61 | \( 1 - 2.37T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 8.62T + 73T^{2} \) |
| 79 | \( 1 - 6.38T + 79T^{2} \) |
| 83 | \( 1 - 0.980T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−7.966529819936349742408078806834, −7.23142840286453525820039159077, −6.50448217218303099190085927216, −5.61632188031020243198030614586, −5.45796205795822622215760398796, −4.33101049726707316387900923074, −3.50869133988495628315763914577, −2.98181961160205912482757911584, −2.08956951393323774427269764016, −0.71552216349871334569662790271,
0.71552216349871334569662790271, 2.08956951393323774427269764016, 2.98181961160205912482757911584, 3.50869133988495628315763914577, 4.33101049726707316387900923074, 5.45796205795822622215760398796, 5.61632188031020243198030614586, 6.50448217218303099190085927216, 7.23142840286453525820039159077, 7.966529819936349742408078806834
