| L(s) = 1 | − 1.14·2-s − 2.15·3-s − 0.679·4-s + 3.87·5-s + 2.47·6-s + 3.07·8-s + 1.63·9-s − 4.45·10-s + 11-s + 1.46·12-s − 4.09·13-s − 8.34·15-s − 2.17·16-s + 0.824·17-s − 1.87·18-s − 4.50·19-s − 2.63·20-s − 1.14·22-s + 4.86·23-s − 6.62·24-s + 10.0·25-s + 4.70·26-s + 2.93·27-s + 7.79·29-s + 9.58·30-s − 3.82·31-s − 3.65·32-s + ⋯ |
| L(s) = 1 | − 0.812·2-s − 1.24·3-s − 0.339·4-s + 1.73·5-s + 1.00·6-s + 1.08·8-s + 0.545·9-s − 1.40·10-s + 0.301·11-s + 0.422·12-s − 1.13·13-s − 2.15·15-s − 0.544·16-s + 0.200·17-s − 0.442·18-s − 1.03·19-s − 0.589·20-s − 0.244·22-s + 1.01·23-s − 1.35·24-s + 2.00·25-s + 0.923·26-s + 0.565·27-s + 1.44·29-s + 1.75·30-s − 0.687·31-s − 0.646·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7026456177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7026456177\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 1.14T + 2T^{2} \) |
| 3 | \( 1 + 2.15T + 3T^{2} \) |
| 5 | \( 1 - 3.87T + 5T^{2} \) |
| 13 | \( 1 + 4.09T + 13T^{2} \) |
| 17 | \( 1 - 0.824T + 17T^{2} \) |
| 19 | \( 1 + 4.50T + 19T^{2} \) |
| 23 | \( 1 - 4.86T + 23T^{2} \) |
| 29 | \( 1 - 7.79T + 29T^{2} \) |
| 31 | \( 1 + 3.82T + 31T^{2} \) |
| 37 | \( 1 - 8.11T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 1.86T + 43T^{2} \) |
| 47 | \( 1 + 7.69T + 47T^{2} \) |
| 53 | \( 1 + 3.93T + 53T^{2} \) |
| 59 | \( 1 - 9.45T + 59T^{2} \) |
| 61 | \( 1 - 8.40T + 61T^{2} \) |
| 67 | \( 1 - 9.45T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 6.19T + 73T^{2} \) |
| 79 | \( 1 - 0.868T + 79T^{2} \) |
| 83 | \( 1 + 8.19T + 83T^{2} \) |
| 89 | \( 1 - 3.87T + 89T^{2} \) |
| 97 | \( 1 - 2.10T + 97T^{2} \) |
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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
−10.59618260365424997500550565811, −9.895910356994280285571320723162, −9.371801136994302719480292590169, −8.385936123226107691204226637934, −7.00402362064669368828621007556, −6.23914089943142914675970418024, −5.28848266689762919757698961120, −4.62690215342762026147858277745, −2.36545606481178679314379470923, −0.934402175737401977273665877553,
0.934402175737401977273665877553, 2.36545606481178679314379470923, 4.62690215342762026147858277745, 5.28848266689762919757698961120, 6.23914089943142914675970418024, 7.00402362064669368828621007556, 8.385936123226107691204226637934, 9.371801136994302719480292590169, 9.895910356994280285571320723162, 10.59618260365424997500550565811
