| L(s) = 1 | + 1.41·2-s − 1.41·3-s + 0.585·5-s − 2.00·6-s − 0.585·7-s − 2.82·8-s − 0.999·9-s + 0.828·10-s + 1.82·11-s + 3.82·13-s − 0.828·14-s − 0.828·15-s − 4.00·16-s + 7.82·17-s − 1.41·18-s − 4.82·19-s + 0.828·21-s + 2.58·22-s − 4.65·23-s + 4·24-s − 4.65·25-s + 5.41·26-s + 5.65·27-s + 4.24·29-s − 1.17·30-s + ⋯ |
| L(s) = 1 | + 1.00·2-s − 0.816·3-s + 0.261·5-s − 0.816·6-s − 0.221·7-s − 0.999·8-s − 0.333·9-s + 0.261·10-s + 0.551·11-s + 1.06·13-s − 0.221·14-s − 0.213·15-s − 1.00·16-s + 1.89·17-s − 0.333·18-s − 1.10·19-s + 0.180·21-s + 0.551·22-s − 0.971·23-s + 0.816·24-s − 0.931·25-s + 1.06·26-s + 1.08·27-s + 0.787·29-s − 0.213·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9213280172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9213280172\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 - T \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 - 0.585T + 5T^{2} \) |
| 7 | \( 1 + 0.585T + 7T^{2} \) |
| 11 | \( 1 - 1.82T + 11T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 17 | \( 1 - 7.82T + 17T^{2} \) |
| 19 | \( 1 + 4.82T + 19T^{2} \) |
| 23 | \( 1 + 4.65T + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 + 3.82T + 41T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 8.17T + 53T^{2} \) |
| 59 | \( 1 - 0.828T + 59T^{2} \) |
| 61 | \( 1 - 8.24T + 61T^{2} \) |
| 67 | \( 1 - 9.48T + 67T^{2} \) |
| 71 | \( 1 + 8.82T + 71T^{2} \) |
| 73 | \( 1 + 7.75T + 73T^{2} \) |
| 79 | \( 1 + 0.828T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 + 3.82T + 97T^{2} \) |
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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
−15.96862753327548951657322812928, −14.54902643044876883916524676202, −13.75477488135291936020076938368, −12.45506562106852186119984380868, −11.68335522144352399410685135764, −10.18631591820728050445314586438, −8.587071820675756351197364898354, −6.30292008875009122275168767003, −5.49340378863976578926921028119, −3.72574111701961389767509613392,
3.72574111701961389767509613392, 5.49340378863976578926921028119, 6.30292008875009122275168767003, 8.587071820675756351197364898354, 10.18631591820728050445314586438, 11.68335522144352399410685135764, 12.45506562106852186119984380868, 13.75477488135291936020076938368, 14.54902643044876883916524676202, 15.96862753327548951657322812928
