L(s) = 1 | − 1.61·2-s + 2.78·3-s + 0.616·4-s + 5-s − 4.50·6-s − 5.17·7-s + 2.23·8-s + 4.74·9-s − 1.61·10-s − 11-s + 1.71·12-s − 1.96·13-s + 8.36·14-s + 2.78·15-s − 4.85·16-s − 1.20·17-s − 7.67·18-s + 1.46·19-s + 0.616·20-s − 14.3·21-s + 1.61·22-s + 5.25·23-s + 6.22·24-s + 25-s + 3.17·26-s + 4.85·27-s − 3.18·28-s + ⋯ |
L(s) = 1 | − 1.14·2-s + 1.60·3-s + 0.308·4-s + 0.447·5-s − 1.83·6-s − 1.95·7-s + 0.791·8-s + 1.58·9-s − 0.511·10-s − 0.301·11-s + 0.495·12-s − 0.544·13-s + 2.23·14-s + 0.718·15-s − 1.21·16-s − 0.292·17-s − 1.80·18-s + 0.336·19-s + 0.137·20-s − 3.14·21-s + 0.344·22-s + 1.09·23-s + 1.27·24-s + 0.200·25-s + 0.623·26-s + 0.933·27-s − 0.602·28-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
\(L(1)\) |
\(\approx\) |
\(1.348542649\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.348542649\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 - 2.78T + 3T^{2} \) |
| 7 | \( 1 + 5.17T + 7T^{2} \) |
| 13 | \( 1 + 1.96T + 13T^{2} \) |
| 17 | \( 1 + 1.20T + 17T^{2} \) |
| 19 | \( 1 - 1.46T + 19T^{2} \) |
| 23 | \( 1 - 5.25T + 23T^{2} \) |
| 29 | \( 1 + 3.23T + 29T^{2} \) |
| 31 | \( 1 - 4.38T + 31T^{2} \) |
| 37 | \( 1 - 1.24T + 37T^{2} \) |
| 41 | \( 1 - 0.338T + 41T^{2} \) |
| 43 | \( 1 - 0.413T + 43T^{2} \) |
| 47 | \( 1 + 3.40T + 47T^{2} \) |
| 53 | \( 1 - 5.96T + 53T^{2} \) |
| 59 | \( 1 - 7.57T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 2.61T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 79 | \( 1 + 4.74T + 79T^{2} \) |
| 83 | \( 1 - 9.19T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−8.678457616936963244711762551449, −7.933778166939759377738534730892, −7.10053193386398326607472128418, −6.79452718309227187330024536626, −5.59582379171175880290872405532, −4.42382933563384163651949291738, −3.47528949969350604483224715668, −2.79452543674253276797004643340, −2.09755671559730001961350261888, −0.71260141339860930766701241625,
0.71260141339860930766701241625, 2.09755671559730001961350261888, 2.79452543674253276797004643340, 3.47528949969350604483224715668, 4.42382933563384163651949291738, 5.59582379171175880290872405532, 6.79452718309227187330024536626, 7.10053193386398326607472128418, 7.933778166939759377738534730892, 8.678457616936963244711762551449