| L(s) = 1 | + 3-s − 3.92·5-s − 3.57·7-s + 9-s + 11-s − 13-s − 3.92·15-s + 0.349·17-s + 0.532·19-s − 3.57·21-s + 5.67·23-s + 10.4·25-s + 27-s + 2.55·29-s − 1.81·31-s + 33-s + 14.0·35-s + 3.76·37-s − 39-s + 5.43·41-s − 0.696·43-s − 3.92·45-s + 1.06·47-s + 5.81·49-s + 0.349·51-s − 2·53-s − 3.92·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.75·5-s − 1.35·7-s + 0.333·9-s + 0.301·11-s − 0.277·13-s − 1.01·15-s + 0.0848·17-s + 0.122·19-s − 0.781·21-s + 1.18·23-s + 2.08·25-s + 0.192·27-s + 0.474·29-s − 0.326·31-s + 0.174·33-s + 2.37·35-s + 0.618·37-s − 0.160·39-s + 0.849·41-s − 0.106·43-s − 0.585·45-s + 0.155·47-s + 0.830·49-s + 0.0489·51-s − 0.274·53-s − 0.529·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 3.92T + 5T^{2} \) |
| 7 | \( 1 + 3.57T + 7T^{2} \) |
| 17 | \( 1 - 0.349T + 17T^{2} \) |
| 19 | \( 1 - 0.532T + 19T^{2} \) |
| 23 | \( 1 - 5.67T + 23T^{2} \) |
| 29 | \( 1 - 2.55T + 29T^{2} \) |
| 31 | \( 1 + 1.81T + 31T^{2} \) |
| 37 | \( 1 - 3.76T + 37T^{2} \) |
| 41 | \( 1 - 5.43T + 41T^{2} \) |
| 43 | \( 1 + 0.696T + 43T^{2} \) |
| 47 | \( 1 - 1.06T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 4.11T + 59T^{2} \) |
| 61 | \( 1 - 3.13T + 61T^{2} \) |
| 67 | \( 1 - 8.02T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 1.65T + 79T^{2} \) |
| 83 | \( 1 + 4.22T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 - 3.76T + 97T^{2} \) |
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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
−7.57067747604185637345124961839, −7.03879697239856799605374012425, −6.52199138033617803218836887949, −5.43997536194154430873741304314, −4.41134592296333380276876595113, −3.93496227636032579774937258761, −3.14360197769000286279902462798, −2.73178130994908457143786968496, −1.06214217381596443893737668148, 0,
1.06214217381596443893737668148, 2.73178130994908457143786968496, 3.14360197769000286279902462798, 3.93496227636032579774937258761, 4.41134592296333380276876595113, 5.43997536194154430873741304314, 6.52199138033617803218836887949, 7.03879697239856799605374012425, 7.57067747604185637345124961839
