| L(s) = 1 | + 2.09·2-s − 2.95·3-s + 2.40·4-s + 2.07·5-s − 6.20·6-s − 4.57·7-s + 0.843·8-s + 5.74·9-s + 4.35·10-s − 3.38·11-s − 7.10·12-s + 5.90·13-s − 9.59·14-s − 6.14·15-s − 3.03·16-s − 0.0690·17-s + 12.0·18-s − 19-s + 4.98·20-s + 13.5·21-s − 7.10·22-s − 5.51·23-s − 2.49·24-s − 0.688·25-s + 12.3·26-s − 8.11·27-s − 10.9·28-s + ⋯ |
| L(s) = 1 | + 1.48·2-s − 1.70·3-s + 1.20·4-s + 0.928·5-s − 2.53·6-s − 1.72·7-s + 0.298·8-s + 1.91·9-s + 1.37·10-s − 1.02·11-s − 2.05·12-s + 1.63·13-s − 2.56·14-s − 1.58·15-s − 0.758·16-s − 0.0167·17-s + 2.84·18-s − 0.229·19-s + 1.11·20-s + 2.95·21-s − 1.51·22-s − 1.14·23-s − 0.509·24-s − 0.137·25-s + 2.42·26-s − 1.56·27-s − 2.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.779047578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.779047578\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + T \) |
| 317 | \( 1 - T \) |
| good | 2 | \( 1 - 2.09T + 2T^{2} \) |
| 3 | \( 1 + 2.95T + 3T^{2} \) |
| 5 | \( 1 - 2.07T + 5T^{2} \) |
| 7 | \( 1 + 4.57T + 7T^{2} \) |
| 11 | \( 1 + 3.38T + 11T^{2} \) |
| 13 | \( 1 - 5.90T + 13T^{2} \) |
| 17 | \( 1 + 0.0690T + 17T^{2} \) |
| 23 | \( 1 + 5.51T + 23T^{2} \) |
| 29 | \( 1 + 4.92T + 29T^{2} \) |
| 31 | \( 1 - 8.74T + 31T^{2} \) |
| 37 | \( 1 - 3.85T + 37T^{2} \) |
| 41 | \( 1 - 5.50T + 41T^{2} \) |
| 43 | \( 1 + 3.35T + 43T^{2} \) |
| 47 | \( 1 - 8.58T + 47T^{2} \) |
| 53 | \( 1 + 8.61T + 53T^{2} \) |
| 59 | \( 1 + 8.82T + 59T^{2} \) |
| 61 | \( 1 - 7.51T + 61T^{2} \) |
| 67 | \( 1 + 7.81T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 6.83T + 79T^{2} \) |
| 83 | \( 1 + 2.21T + 83T^{2} \) |
| 89 | \( 1 + 2.64T + 89T^{2} \) |
| 97 | \( 1 - 6.83T + 97T^{2} \) |
| 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
−7.70197142268679217343583589220, −6.61456395868912684278976354669, −6.27539905886773555012844650047, −5.88490669850673478105878371052, −5.53492050024352364595546512900, −4.55641414445808286537349636842, −3.86765333900640662055572097499, −3.05012294332422656744284785812, −2.05282141769219586766801429395, −0.58396556575641600219798947476,
0.58396556575641600219798947476, 2.05282141769219586766801429395, 3.05012294332422656744284785812, 3.86765333900640662055572097499, 4.55641414445808286537349636842, 5.53492050024352364595546512900, 5.88490669850673478105878371052, 6.27539905886773555012844650047, 6.61456395868912684278976354669, 7.70197142268679217343583589220
