L(s) = 1 | − 0.0798·2-s + 0.0600·3-s − 1.99·4-s + 5-s − 0.00480·6-s − 4.79·7-s + 0.319·8-s − 2.99·9-s − 0.0798·10-s − 11-s − 0.119·12-s − 6.38·13-s + 0.382·14-s + 0.0600·15-s + 3.96·16-s − 1.67·17-s + 0.239·18-s − 7.70·19-s − 1.99·20-s − 0.287·21-s + 0.0798·22-s − 6.03·23-s + 0.0191·24-s + 25-s + 0.510·26-s − 0.360·27-s + 9.55·28-s + ⋯ |
L(s) = 1 | − 0.0564·2-s + 0.0346·3-s − 0.996·4-s + 0.447·5-s − 0.00195·6-s − 1.81·7-s + 0.112·8-s − 0.998·9-s − 0.0252·10-s − 0.301·11-s − 0.0345·12-s − 1.77·13-s + 0.102·14-s + 0.0155·15-s + 0.990·16-s − 0.406·17-s + 0.0564·18-s − 1.76·19-s − 0.445·20-s − 0.0628·21-s + 0.0170·22-s − 1.25·23-s + 0.00391·24-s + 0.200·25-s + 0.100·26-s − 0.0693·27-s + 1.80·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.01388542993\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01388542993\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
good | 2 | \( 1 + 0.0798T + 2T^{2} \) |
| 3 | \( 1 - 0.0600T + 3T^{2} \) |
| 7 | \( 1 + 4.79T + 7T^{2} \) |
| 13 | \( 1 + 6.38T + 13T^{2} \) |
| 17 | \( 1 + 1.67T + 17T^{2} \) |
| 19 | \( 1 + 7.70T + 19T^{2} \) |
| 23 | \( 1 + 6.03T + 23T^{2} \) |
| 29 | \( 1 - 4.87T + 29T^{2} \) |
| 31 | \( 1 + 9.07T + 31T^{2} \) |
| 37 | \( 1 + 0.534T + 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 + 2.67T + 43T^{2} \) |
| 47 | \( 1 + 9.73T + 47T^{2} \) |
| 53 | \( 1 + 2.69T + 53T^{2} \) |
| 59 | \( 1 + 3.42T + 59T^{2} \) |
| 61 | \( 1 + 2.63T + 61T^{2} \) |
| 67 | \( 1 - 9.96T + 67T^{2} \) |
| 71 | \( 1 - 0.362T + 71T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 1.49T + 83T^{2} \) |
| 89 | \( 1 + 8.90T + 89T^{2} \) |
| 97 | \( 1 + 6.72T + 97T^{2} \) |
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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
−8.597078189147951041472672141740, −7.84117125663288995201652430352, −6.86059872867176542919962987977, −6.15458754300707278114950563736, −5.57425096543803460667683479427, −4.65951791880265771123668639162, −3.85876795950928268027779223241, −2.89076829840303697775032187557, −2.23462943073437082739022383287, −0.06214479488649982840432657144,
0.06214479488649982840432657144, 2.23462943073437082739022383287, 2.89076829840303697775032187557, 3.85876795950928268027779223241, 4.65951791880265771123668639162, 5.57425096543803460667683479427, 6.15458754300707278114950563736, 6.86059872867176542919962987977, 7.84117125663288995201652430352, 8.597078189147951041472672141740