| L(s) = 1 | + 1.41·2-s − 3.41·5-s − 3.41·7-s − 2.82·8-s − 4.82·10-s − 2.41·11-s + 6.24·13-s − 4.82·14-s − 4.00·16-s + 0.414·17-s − 5.41·19-s − 3.41·22-s + 1.41·23-s + 6.65·25-s + 8.82·26-s + 6.07·29-s − 3·31-s + 0.585·34-s + 11.6·35-s − 9.48·37-s − 7.65·38-s + 9.65·40-s + 41-s − 5·43-s + ⋯ |
| L(s) = 1 | + 1.00·2-s − 1.52·5-s − 1.29·7-s − 0.999·8-s − 1.52·10-s − 0.727·11-s + 1.73·13-s − 1.29·14-s − 1.00·16-s + 0.100·17-s − 1.24·19-s − 0.727·22-s + 0.294·23-s + 1.33·25-s + 1.73·26-s + 1.12·29-s − 0.538·31-s + 0.100·34-s + 1.97·35-s − 1.55·37-s − 1.24·38-s + 1.52·40-s + 0.156·41-s − 0.762·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 5 | \( 1 + 3.41T + 5T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 + 2.41T + 11T^{2} \) |
| 13 | \( 1 - 6.24T + 13T^{2} \) |
| 17 | \( 1 - 0.414T + 17T^{2} \) |
| 19 | \( 1 + 5.41T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 6.07T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 9.48T + 37T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 6.82T + 53T^{2} \) |
| 59 | \( 1 - 8.48T + 59T^{2} \) |
| 61 | \( 1 + 4.65T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 + 7.65T + 79T^{2} \) |
| 83 | \( 1 + 1.07T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 7.75T + 97T^{2} \) |
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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
−11.18801419592298282826972893853, −10.24292062051570146064374792532, −8.824007390134708758750617819930, −8.272920772316086025732140442980, −6.84636087053774927908245287045, −6.07520995616118819909379070769, −4.74560331663843961833492597072, −3.70061096793759382935361434051, −3.19268338333349834296604449367, 0,
3.19268338333349834296604449367, 3.70061096793759382935361434051, 4.74560331663843961833492597072, 6.07520995616118819909379070769, 6.84636087053774927908245287045, 8.272920772316086025732140442980, 8.824007390134708758750617819930, 10.24292062051570146064374792532, 11.18801419592298282826972893853
