L(s) = 1 | − 0.618·2-s − 1.61·4-s − 5-s − 7-s + 2.23·8-s + 0.618·10-s − 3.47·13-s + 0.618·14-s + 1.85·16-s + 5.23·17-s + 6.70·19-s + 1.61·20-s − 5.70·23-s − 4·25-s + 2.14·26-s + 1.61·28-s + 5·29-s − 5.23·31-s − 5.61·32-s − 3.23·34-s + 35-s − 7·37-s − 4.14·38-s − 2.23·40-s − 2.47·41-s − 5.70·43-s + 3.52·46-s + ⋯ |
L(s) = 1 | − 0.437·2-s − 0.809·4-s − 0.447·5-s − 0.377·7-s + 0.790·8-s + 0.195·10-s − 0.962·13-s + 0.165·14-s + 0.463·16-s + 1.26·17-s + 1.53·19-s + 0.361·20-s − 1.19·23-s − 0.800·25-s + 0.420·26-s + 0.305·28-s + 0.928·29-s − 0.940·31-s − 0.993·32-s − 0.554·34-s + 0.169·35-s − 1.15·37-s − 0.672·38-s − 0.353·40-s − 0.386·41-s − 0.870·43-s + 0.520·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 13 | \( 1 + 3.47T + 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 5.23T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 + 0.236T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 9.76T + 67T^{2} \) |
| 71 | \( 1 - 2.47T + 71T^{2} \) |
| 73 | \( 1 + 4.52T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 6.76T + 83T^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 - 9.70T + 97T^{2} \) |
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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
−7.50681152914112853247300145288, −7.25545466232050191809061493116, −6.04751281375309092309197997169, −5.30193137437481278032074178052, −4.82386464855389994609493442070, −3.69881020298103924638242649404, −3.43304295095258331453813440752, −2.13247712610977997763890786436, −0.995850504174931781794216194055, 0,
0.995850504174931781794216194055, 2.13247712610977997763890786436, 3.43304295095258331453813440752, 3.69881020298103924638242649404, 4.82386464855389994609493442070, 5.30193137437481278032074178052, 6.04751281375309092309197997169, 7.25545466232050191809061493116, 7.50681152914112853247300145288