| L(s) = 1 | + 2.41·2-s + 3-s + 3.82·4-s + 2.41·6-s − 0.828·7-s + 4.41·8-s + 9-s − 11-s + 3.82·12-s + 5.65·13-s − 1.99·14-s + 2.99·16-s + 1.17·17-s + 2.41·18-s − 6.82·19-s − 0.828·21-s − 2.41·22-s + 4·23-s + 4.41·24-s + 13.6·26-s + 27-s − 3.17·28-s − 4.82·29-s − 1.58·32-s − 33-s + 2.82·34-s + 3.82·36-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 0.577·3-s + 1.91·4-s + 0.985·6-s − 0.313·7-s + 1.56·8-s + 0.333·9-s − 0.301·11-s + 1.10·12-s + 1.56·13-s − 0.534·14-s + 0.749·16-s + 0.284·17-s + 0.569·18-s − 1.56·19-s − 0.180·21-s − 0.514·22-s + 0.834·23-s + 0.901·24-s + 2.67·26-s + 0.192·27-s − 0.599·28-s − 0.896·29-s − 0.280·32-s − 0.174·33-s + 0.485·34-s + 0.638·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.777599497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.777599497\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 7 | \( 1 + 0.828T + 7T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 - 1.17T + 17T^{2} \) |
| 19 | \( 1 + 6.82T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 - 4.82T + 41T^{2} \) |
| 43 | \( 1 - 8.82T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 9.31T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 - 2.34T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 - 10T + 83T^{2} \) |
| 89 | \( 1 - 3.65T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−10.82002632252892950333455169568, −9.286489844669918528238429666743, −8.480940432247750198962491246559, −7.37942904110561407914367273826, −6.43228708704950001734148242244, −5.80190799950083295532458278252, −4.67139490217686918531454533113, −3.78188653335801157699735763598, −3.08785192459668953939314963464, −1.86502780182333678808141367125,
1.86502780182333678808141367125, 3.08785192459668953939314963464, 3.78188653335801157699735763598, 4.67139490217686918531454533113, 5.80190799950083295532458278252, 6.43228708704950001734148242244, 7.37942904110561407914367273826, 8.480940432247750198962491246559, 9.286489844669918528238429666743, 10.82002632252892950333455169568
