L(s) = 1 | − 2.06·2-s − 3-s + 2.27·4-s + 5-s + 2.06·6-s + 7-s − 0.558·8-s + 9-s − 2.06·10-s + 11-s − 2.27·12-s + 0.491·13-s − 2.06·14-s − 15-s − 3.38·16-s + 7.40·17-s − 2.06·18-s − 0.729·19-s + 2.27·20-s − 21-s − 2.06·22-s + 8.92·23-s + 0.558·24-s + 25-s − 1.01·26-s − 27-s + 2.27·28-s + ⋯ |
L(s) = 1 | − 1.46·2-s − 0.577·3-s + 1.13·4-s + 0.447·5-s + 0.843·6-s + 0.377·7-s − 0.197·8-s + 0.333·9-s − 0.653·10-s + 0.301·11-s − 0.655·12-s + 0.136·13-s − 0.552·14-s − 0.258·15-s − 0.846·16-s + 1.79·17-s − 0.487·18-s − 0.167·19-s + 0.507·20-s − 0.218·21-s − 0.440·22-s + 1.86·23-s + 0.113·24-s + 0.200·25-s − 0.199·26-s − 0.192·27-s + 0.429·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7770365689\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7770365689\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + 2.06T + 2T^{2} \) |
| 13 | \( 1 - 0.491T + 13T^{2} \) |
| 17 | \( 1 - 7.40T + 17T^{2} \) |
| 19 | \( 1 + 0.729T + 19T^{2} \) |
| 23 | \( 1 - 8.92T + 23T^{2} \) |
| 29 | \( 1 + 4.91T + 29T^{2} \) |
| 31 | \( 1 + 7.64T + 31T^{2} \) |
| 37 | \( 1 - 7.10T + 37T^{2} \) |
| 41 | \( 1 + 2.04T + 41T^{2} \) |
| 43 | \( 1 + 9.55T + 43T^{2} \) |
| 47 | \( 1 + 6.68T + 47T^{2} \) |
| 53 | \( 1 + 0.862T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 - 4.86T + 61T^{2} \) |
| 67 | \( 1 + 3.49T + 67T^{2} \) |
| 71 | \( 1 + 6.18T + 71T^{2} \) |
| 73 | \( 1 + 0.132T + 73T^{2} \) |
| 79 | \( 1 - 9.77T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 0.745T + 89T^{2} \) |
| 97 | \( 1 + 3.76T + 97T^{2} \) |
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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
−9.712339274809892323149177082948, −9.139810824702941942489327940494, −8.237624238763816342119247260401, −7.44285756760641637552437773296, −6.75769397514392743183776224517, −5.65881203589831497196727500984, −4.87294389213988524812543160591, −3.39894471799193583221846511621, −1.83352336461956567819928324923, −0.908877128149224408316155164361,
0.908877128149224408316155164361, 1.83352336461956567819928324923, 3.39894471799193583221846511621, 4.87294389213988524812543160591, 5.65881203589831497196727500984, 6.75769397514392743183776224517, 7.44285756760641637552437773296, 8.237624238763816342119247260401, 9.139810824702941942489327940494, 9.712339274809892323149177082948