| L(s) = 1 | − 3-s + 1.25·5-s − 4.98·7-s + 9-s − 0.651·11-s − 2.98·13-s − 1.25·15-s − 0.150·17-s − 4.18·19-s + 4.98·21-s + 6.72·23-s − 3.43·25-s − 27-s + 5.88·29-s + 8.31·31-s + 0.651·33-s − 6.24·35-s + 9.27·37-s + 2.98·39-s − 41-s + 8.57·43-s + 1.25·45-s − 5.08·47-s + 17.8·49-s + 0.150·51-s + 6.54·53-s − 0.814·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.559·5-s − 1.88·7-s + 0.333·9-s − 0.196·11-s − 0.829·13-s − 0.322·15-s − 0.0364·17-s − 0.960·19-s + 1.08·21-s + 1.40·23-s − 0.687·25-s − 0.192·27-s + 1.09·29-s + 1.49·31-s + 0.113·33-s − 1.05·35-s + 1.52·37-s + 0.478·39-s − 0.156·41-s + 1.30·43-s + 0.186·45-s − 0.742·47-s + 2.55·49-s + 0.0210·51-s + 0.898·53-s − 0.109·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| good | 5 | \( 1 - 1.25T + 5T^{2} \) |
| 7 | \( 1 + 4.98T + 7T^{2} \) |
| 11 | \( 1 + 0.651T + 11T^{2} \) |
| 13 | \( 1 + 2.98T + 13T^{2} \) |
| 17 | \( 1 + 0.150T + 17T^{2} \) |
| 19 | \( 1 + 4.18T + 19T^{2} \) |
| 23 | \( 1 - 6.72T + 23T^{2} \) |
| 29 | \( 1 - 5.88T + 29T^{2} \) |
| 31 | \( 1 - 8.31T + 31T^{2} \) |
| 37 | \( 1 - 9.27T + 37T^{2} \) |
| 43 | \( 1 - 8.57T + 43T^{2} \) |
| 47 | \( 1 + 5.08T + 47T^{2} \) |
| 53 | \( 1 - 6.54T + 53T^{2} \) |
| 59 | \( 1 + 2.69T + 59T^{2} \) |
| 61 | \( 1 - 4.44T + 61T^{2} \) |
| 67 | \( 1 - 2.39T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 - 9.11T + 73T^{2} \) |
| 79 | \( 1 + 9.17T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 5.39T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−7.20009021201026215877309640255, −6.70877743649351271394619611888, −6.15374587010574862742737709216, −5.62627705091182523877339813250, −4.67280651573857327219625303955, −4.02080201915204982079845093671, −2.80929415457418482462812960469, −2.57564118757510397892265855360, −1.02556476546399615375478385308, 0,
1.02556476546399615375478385308, 2.57564118757510397892265855360, 2.80929415457418482462812960469, 4.02080201915204982079845093671, 4.67280651573857327219625303955, 5.62627705091182523877339813250, 6.15374587010574862742737709216, 6.70877743649351271394619611888, 7.20009021201026215877309640255
