L(s) = 1 | − 2-s + 1.14·3-s + 4-s + 0.899·5-s − 1.14·6-s + 3.82·7-s − 8-s − 1.67·9-s − 0.899·10-s − 1.31·11-s + 1.14·12-s + 7.00·13-s − 3.82·14-s + 1.03·15-s + 16-s + 5.78·17-s + 1.67·18-s − 19-s + 0.899·20-s + 4.40·21-s + 1.31·22-s + 7.02·23-s − 1.14·24-s − 4.19·25-s − 7.00·26-s − 5.37·27-s + 3.82·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.663·3-s + 0.5·4-s + 0.402·5-s − 0.469·6-s + 1.44·7-s − 0.353·8-s − 0.559·9-s − 0.284·10-s − 0.396·11-s + 0.331·12-s + 1.94·13-s − 1.02·14-s + 0.267·15-s + 0.250·16-s + 1.40·17-s + 0.395·18-s − 0.229·19-s + 0.201·20-s + 0.960·21-s + 0.280·22-s + 1.46·23-s − 0.234·24-s − 0.838·25-s − 1.37·26-s − 1.03·27-s + 0.723·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.811084735\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.811084735\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
good | 3 | \( 1 - 1.14T + 3T^{2} \) |
| 5 | \( 1 - 0.899T + 5T^{2} \) |
| 7 | \( 1 - 3.82T + 7T^{2} \) |
| 11 | \( 1 + 1.31T + 11T^{2} \) |
| 13 | \( 1 - 7.00T + 13T^{2} \) |
| 17 | \( 1 - 5.78T + 17T^{2} \) |
| 23 | \( 1 - 7.02T + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 - 6.72T + 31T^{2} \) |
| 37 | \( 1 - 6.42T + 37T^{2} \) |
| 41 | \( 1 + 1.82T + 41T^{2} \) |
| 43 | \( 1 - 0.621T + 43T^{2} \) |
| 47 | \( 1 + 1.08T + 47T^{2} \) |
| 53 | \( 1 - 8.75T + 53T^{2} \) |
| 59 | \( 1 + 2.72T + 59T^{2} \) |
| 61 | \( 1 - 0.0792T + 61T^{2} \) |
| 67 | \( 1 - 6.16T + 67T^{2} \) |
| 71 | \( 1 - 5.66T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 5.88T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + 7.00T + 97T^{2} \) |
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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
−8.089461385941584489938194883781, −7.52665536033193129422101738125, −6.50536138254374214476549226415, −5.61240955326190815200874351656, −5.38467021934766026364084096908, −4.09404796463193543571131807258, −3.37372497271410893445168290881, −2.51043007145034390752947865115, −1.64743301050634828148105024949, −0.979920538882710360771952521731,
0.979920538882710360771952521731, 1.64743301050634828148105024949, 2.51043007145034390752947865115, 3.37372497271410893445168290881, 4.09404796463193543571131807258, 5.38467021934766026364084096908, 5.61240955326190815200874351656, 6.50536138254374214476549226415, 7.52665536033193129422101738125, 8.089461385941584489938194883781