| L(s) = 1 | + 15·5-s + 25·7-s + 15·11-s + 20·13-s + 72·17-s − 2·19-s − 114·23-s + 100·25-s + 30·29-s − 101·31-s + 375·35-s − 430·37-s − 30·41-s − 110·43-s + 330·47-s + 282·49-s + 621·53-s + 225·55-s + 660·59-s − 376·61-s + 300·65-s + 250·67-s + 360·71-s + 785·73-s + 375·77-s − 488·79-s − 489·83-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 1.34·7-s + 0.411·11-s + 0.426·13-s + 1.02·17-s − 0.0241·19-s − 1.03·23-s + 4/5·25-s + 0.192·29-s − 0.585·31-s + 1.81·35-s − 1.91·37-s − 0.114·41-s − 0.390·43-s + 1.02·47-s + 0.822·49-s + 1.60·53-s + 0.551·55-s + 1.45·59-s − 0.789·61-s + 0.572·65-s + 0.455·67-s + 0.601·71-s + 1.25·73-s + 0.555·77-s − 0.694·79-s − 0.646·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.050372157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.050372157\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 7 | \( 1 - 25 T + p^{3} T^{2} \) |
| 11 | \( 1 - 15 T + p^{3} T^{2} \) |
| 13 | \( 1 - 20 T + p^{3} T^{2} \) |
| 17 | \( 1 - 72 T + p^{3} T^{2} \) |
| 19 | \( 1 + 2 T + p^{3} T^{2} \) |
| 23 | \( 1 + 114 T + p^{3} T^{2} \) |
| 29 | \( 1 - 30 T + p^{3} T^{2} \) |
| 31 | \( 1 + 101 T + p^{3} T^{2} \) |
| 37 | \( 1 + 430 T + p^{3} T^{2} \) |
| 41 | \( 1 + 30 T + p^{3} T^{2} \) |
| 43 | \( 1 + 110 T + p^{3} T^{2} \) |
| 47 | \( 1 - 330 T + p^{3} T^{2} \) |
| 53 | \( 1 - 621 T + p^{3} T^{2} \) |
| 59 | \( 1 - 660 T + p^{3} T^{2} \) |
| 61 | \( 1 + 376 T + p^{3} T^{2} \) |
| 67 | \( 1 - 250 T + p^{3} T^{2} \) |
| 71 | \( 1 - 360 T + p^{3} T^{2} \) |
| 73 | \( 1 - 785 T + p^{3} T^{2} \) |
| 79 | \( 1 + 488 T + p^{3} T^{2} \) |
| 83 | \( 1 + 489 T + p^{3} T^{2} \) |
| 89 | \( 1 + 450 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1105 T + p^{3} T^{2} \) |
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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
−10.59199335777222656212673297287, −9.944550218558164504304461614812, −8.904790318448454321747239782596, −8.115474735284305216473207260849, −6.96686166741679146077798305058, −5.78352649374207826925743295618, −5.19845343951551946234578824278, −3.82763111411129667701989521839, −2.15762877241039560527557218277, −1.30551836432768445183321594073,
1.30551836432768445183321594073, 2.15762877241039560527557218277, 3.82763111411129667701989521839, 5.19845343951551946234578824278, 5.78352649374207826925743295618, 6.96686166741679146077798305058, 8.115474735284305216473207260849, 8.904790318448454321747239782596, 9.944550218558164504304461614812, 10.59199335777222656212673297287
