| L(s) = 1 | + 1.41·2-s − 5-s + 1.41·7-s − 2.82·8-s − 1.41·10-s − 11-s + 13-s + 2.00·14-s − 4.00·16-s − 3·19-s − 1.41·22-s + 5·23-s − 4·25-s + 1.41·26-s + 8·29-s − 5.65·31-s − 1.41·35-s − 1.41·37-s − 4.24·38-s + 2.82·40-s − 11·41-s − 9·43-s + 7.07·46-s − 8.48·47-s + ⋯ |
| L(s) = 1 | + 1.00·2-s − 0.447·5-s + 0.534·7-s − 0.999·8-s − 0.447·10-s − 0.301·11-s + 0.277·13-s + 0.534·14-s − 1.00·16-s − 0.688·19-s − 0.301·22-s + 1.04·23-s − 0.800·25-s + 0.277·26-s + 1.48·29-s − 1.01·31-s − 0.239·35-s − 0.232·37-s − 0.688·38-s + 0.447·40-s − 1.71·41-s − 1.37·43-s + 1.04·46-s − 1.23·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 - 5T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 41 | \( 1 + 11T + 41T^{2} \) |
| 43 | \( 1 + 9T + 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 + 2.82T + 53T^{2} \) |
| 59 | \( 1 + 9.89T + 59T^{2} \) |
| 61 | \( 1 + 7.07T + 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 - 7.07T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 4.24T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−8.457082831516044211034578429793, −7.79691265281051227930607866587, −6.72622308619066112764533449728, −6.10984218965192148580390478757, −4.92698454103035525172474583499, −4.81038919621767539359377199819, −3.65698459889650032998969921090, −3.05128961370685963844137340381, −1.72016940680890540653911574939, 0,
1.72016940680890540653911574939, 3.05128961370685963844137340381, 3.65698459889650032998969921090, 4.81038919621767539359377199819, 4.92698454103035525172474583499, 6.10984218965192148580390478757, 6.72622308619066112764533449728, 7.79691265281051227930607866587, 8.457082831516044211034578429793
