| L(s) = 1 | − 0.467·5-s + 3.22·7-s − 3.10·11-s − 2.18·13-s − 3·17-s − 0.0418·19-s + 6.10·23-s − 4.78·25-s − 6.57·29-s + 6.22·31-s − 1.50·35-s + 3.59·37-s − 7.70·41-s + 0.588·43-s − 9.66·47-s + 3.41·49-s − 4.95·53-s + 1.45·55-s − 8.53·59-s − 1.26·61-s + 1.02·65-s − 10.0·67-s + 11.8·71-s − 8.23·73-s − 10.0·77-s − 11.0·79-s − 1.50·83-s + ⋯ |
| L(s) = 1 | − 0.209·5-s + 1.21·7-s − 0.936·11-s − 0.605·13-s − 0.727·17-s − 0.00961·19-s + 1.27·23-s − 0.956·25-s − 1.22·29-s + 1.11·31-s − 0.255·35-s + 0.591·37-s − 1.20·41-s + 0.0897·43-s − 1.40·47-s + 0.487·49-s − 0.681·53-s + 0.195·55-s − 1.11·59-s − 0.162·61-s + 0.126·65-s − 1.22·67-s + 1.40·71-s − 0.963·73-s − 1.14·77-s − 1.24·79-s − 0.165·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 + 0.467T + 5T^{2} \) |
| 7 | \( 1 - 3.22T + 7T^{2} \) |
| 11 | \( 1 + 3.10T + 11T^{2} \) |
| 13 | \( 1 + 2.18T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 0.0418T + 19T^{2} \) |
| 23 | \( 1 - 6.10T + 23T^{2} \) |
| 29 | \( 1 + 6.57T + 29T^{2} \) |
| 31 | \( 1 - 6.22T + 31T^{2} \) |
| 37 | \( 1 - 3.59T + 37T^{2} \) |
| 41 | \( 1 + 7.70T + 41T^{2} \) |
| 43 | \( 1 - 0.588T + 43T^{2} \) |
| 47 | \( 1 + 9.66T + 47T^{2} \) |
| 53 | \( 1 + 4.95T + 53T^{2} \) |
| 59 | \( 1 + 8.53T + 59T^{2} \) |
| 61 | \( 1 + 1.26T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 8.23T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 1.50T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−7.949006234235659012259460467551, −7.61900457199935498004591451961, −6.73255519659357397100650458680, −5.77314750969281902168893971711, −4.85510581554273203279306503146, −4.62926178415234484386524268833, −3.37191862324751162921187324734, −2.41520092911475986425798225864, −1.53061570986950636523135884593, 0,
1.53061570986950636523135884593, 2.41520092911475986425798225864, 3.37191862324751162921187324734, 4.62926178415234484386524268833, 4.85510581554273203279306503146, 5.77314750969281902168893971711, 6.73255519659357397100650458680, 7.61900457199935498004591451961, 7.949006234235659012259460467551
