| L(s) = 1 | − 1.41·3-s − 0.999·9-s − 11-s + 1.41·13-s + 7.07·17-s − 2.82·19-s − 4·23-s − 5·25-s + 5.65·27-s + 2·29-s + 4.24·31-s + 1.41·33-s − 4·37-s − 2.00·39-s − 1.41·41-s + 2·43-s − 9.89·47-s − 10.0·51-s − 4·53-s + 4.00·57-s − 4.24·59-s + 12.7·61-s − 8·67-s + 5.65·69-s − 1.41·73-s + 7.07·75-s − 10·79-s + ⋯ |
| L(s) = 1 | − 0.816·3-s − 0.333·9-s − 0.301·11-s + 0.392·13-s + 1.71·17-s − 0.648·19-s − 0.834·23-s − 25-s + 1.08·27-s + 0.371·29-s + 0.762·31-s + 0.246·33-s − 0.657·37-s − 0.320·39-s − 0.220·41-s + 0.304·43-s − 1.44·47-s − 1.40·51-s − 0.549·53-s + 0.529·57-s − 0.552·59-s + 1.62·61-s − 0.977·67-s + 0.681·69-s − 0.165·73-s + 0.816·75-s − 1.12·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 - 7.07T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 1.41T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 9.89T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + 4.24T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 1.41T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 8.48T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + 8.48T + 97T^{2} \) |
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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
−8.440589346773426646628120495820, −8.079947665412074662667541496867, −7.03501481534130683314167794213, −6.09474967324740660128057137710, −5.67679028033406522090451404935, −4.80664797300500004375766506422, −3.76637679943203824314287076897, −2.78826958433702820106589383271, −1.41410101917632202608535330189, 0,
1.41410101917632202608535330189, 2.78826958433702820106589383271, 3.76637679943203824314287076897, 4.80664797300500004375766506422, 5.67679028033406522090451404935, 6.09474967324740660128057137710, 7.03501481534130683314167794213, 8.079947665412074662667541496867, 8.440589346773426646628120495820
