| L(s) = 1 | − 2·7-s + 4·11-s + 13-s + 4·17-s + 2·19-s − 8·23-s − 2·31-s + 10·37-s − 10·41-s − 3·49-s + 8·53-s − 4·59-s + 2·61-s − 2·67-s − 2·73-s − 8·77-s − 12·83-s − 6·89-s − 2·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 8·119-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 1.20·11-s + 0.277·13-s + 0.970·17-s + 0.458·19-s − 1.66·23-s − 0.359·31-s + 1.64·37-s − 1.56·41-s − 3/7·49-s + 1.09·53-s − 0.520·59-s + 0.256·61-s − 0.244·67-s − 0.234·73-s − 0.911·77-s − 1.31·83-s − 0.635·89-s − 0.209·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.15047414105046, −13.64584565803620, −13.00562779042591, −12.67465388045138, −11.93332989608946, −11.75013459387323, −11.32183488824354, −10.37929111783171, −10.13649292387760, −9.574772956319825, −9.249405980861701, −8.558380272845496, −8.079553944418230, −7.505379039059305, −6.943016688472165, −6.381147571392356, −5.943176003674664, −5.533795516334986, −4.661399715353660, −4.087386107100425, −3.549916265446755, −3.156876288300623, −2.293208218387052, −1.562594253111876, −0.9346901137994854, 0,
0.9346901137994854, 1.562594253111876, 2.293208218387052, 3.156876288300623, 3.549916265446755, 4.087386107100425, 4.661399715353660, 5.533795516334986, 5.943176003674664, 6.381147571392356, 6.943016688472165, 7.505379039059305, 8.079553944418230, 8.558380272845496, 9.249405980861701, 9.574772956319825, 10.13649292387760, 10.37929111783171, 11.32183488824354, 11.75013459387323, 11.93332989608946, 12.67465388045138, 13.00562779042591, 13.64584565803620, 14.15047414105046
