| L(s) = 1 | − 1.41·2-s − 0.414·3-s + 5-s + 0.585·6-s + 2.82·8-s − 2.82·9-s − 1.41·10-s − 0.171·11-s + 4.41·13-s − 0.414·15-s − 4.00·16-s + 3.24·17-s + 4.00·18-s + 6·19-s + 0.242·22-s + 7.41·23-s − 1.17·24-s + 25-s − 6.24·26-s + 2.41·27-s − 8.65·29-s + 0.585·30-s + 10.2·31-s + 0.0710·33-s − 4.58·34-s + ⋯ |
| L(s) = 1 | − 1.00·2-s − 0.239·3-s + 0.447·5-s + 0.239·6-s + 0.999·8-s − 0.942·9-s − 0.447·10-s − 0.0517·11-s + 1.22·13-s − 0.106·15-s − 1.00·16-s + 0.786·17-s + 0.942·18-s + 1.37·19-s + 0.0517·22-s + 1.54·23-s − 0.239·24-s + 0.200·25-s − 1.22·26-s + 0.464·27-s − 1.60·29-s + 0.106·30-s + 1.83·31-s + 0.0123·33-s − 0.786·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7034737732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7034737732\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 3 | \( 1 + 0.414T + 3T^{2} \) |
| 11 | \( 1 + 0.171T + 11T^{2} \) |
| 13 | \( 1 - 4.41T + 13T^{2} \) |
| 17 | \( 1 - 3.24T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 7.41T + 23T^{2} \) |
| 29 | \( 1 + 8.65T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 + 6.24T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 7.24T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 + 2.24T + 59T^{2} \) |
| 61 | \( 1 + 2.82T + 61T^{2} \) |
| 67 | \( 1 + 8.24T + 67T^{2} \) |
| 71 | \( 1 + 3.17T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 - 1.48T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−11.73477395593774816396878547392, −11.00202302594167867959362115395, −10.03940226125010060606135121765, −9.154333733674727999064963391188, −8.411833973854258661558160498613, −7.37480225697226856105903711973, −6.01837265074468781325981483010, −5.00788806556701209132586036557, −3.22276589798217675804356448983, −1.16360101941472512386216658081,
1.16360101941472512386216658081, 3.22276589798217675804356448983, 5.00788806556701209132586036557, 6.01837265074468781325981483010, 7.37480225697226856105903711973, 8.411833973854258661558160498613, 9.154333733674727999064963391188, 10.03940226125010060606135121765, 11.00202302594167867959362115395, 11.73477395593774816396878547392
