L(s) = 1 | − 0.456·2-s − 3-s − 1.79·4-s + 0.456·6-s − 1.73·7-s + 1.73·8-s − 2·9-s − 2.64·11-s + 1.79·12-s + 0.791·14-s + 2.79·16-s − 4.58·17-s + 0.913·18-s + 1.73·19-s + 1.73·21-s + 1.20·22-s + 4.58·23-s − 1.73·24-s + 5·27-s + 3.10·28-s + 4.58·29-s + 6.20·31-s − 4.73·32-s + 2.64·33-s + 2.09·34-s + 3.58·36-s + 7.93·37-s + ⋯ |
L(s) = 1 | − 0.323·2-s − 0.577·3-s − 0.895·4-s + 0.186·6-s − 0.654·7-s + 0.612·8-s − 0.666·9-s − 0.797·11-s + 0.517·12-s + 0.211·14-s + 0.697·16-s − 1.11·17-s + 0.215·18-s + 0.397·19-s + 0.377·21-s + 0.257·22-s + 0.955·23-s − 0.353·24-s + 0.962·27-s + 0.586·28-s + 0.850·29-s + 1.11·31-s − 0.837·32-s + 0.460·33-s + 0.359·34-s + 0.597·36-s + 1.30·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 0.456T + 2T^{2} \) |
| 3 | \( 1 + T + 3T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 11 | \( 1 + 2.64T + 11T^{2} \) |
| 17 | \( 1 + 4.58T + 17T^{2} \) |
| 19 | \( 1 - 1.73T + 19T^{2} \) |
| 23 | \( 1 - 4.58T + 23T^{2} \) |
| 29 | \( 1 - 4.58T + 29T^{2} \) |
| 31 | \( 1 - 6.20T + 31T^{2} \) |
| 37 | \( 1 - 7.93T + 37T^{2} \) |
| 41 | \( 1 + 2.64T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 1.82T + 47T^{2} \) |
| 53 | \( 1 + 7.58T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 + 1.41T + 61T^{2} \) |
| 67 | \( 1 - 1.00T + 67T^{2} \) |
| 71 | \( 1 - 7.02T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 6.01T + 83T^{2} \) |
| 89 | \( 1 + 9.57T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.161767751634403974411486906717, −7.40310772083763345388816023787, −6.40866387285476154558908896391, −5.90464586975854880484947871814, −4.86863507100071047886778358060, −4.58837090644337416749002677574, −3.29244026978057639255724617659, −2.58082788530152075936552590618, −0.949847156161175933242480810665, 0,
0.949847156161175933242480810665, 2.58082788530152075936552590618, 3.29244026978057639255724617659, 4.58837090644337416749002677574, 4.86863507100071047886778358060, 5.90464586975854880484947871814, 6.40866387285476154558908896391, 7.40310772083763345388816023787, 8.161767751634403974411486906717