| L(s) = 1 | + 2·2-s + 5·3-s + 4·4-s + 5·5-s + 10·6-s − 7·7-s + 8·8-s − 2·9-s + 10·10-s − 11-s + 20·12-s + 7·13-s − 14·14-s + 25·15-s + 16·16-s − 51·17-s − 4·18-s + 30·19-s + 20·20-s − 35·21-s − 2·22-s − 50·23-s + 40·24-s + 25·25-s + 14·26-s − 145·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.962·3-s + 1/2·4-s + 0.447·5-s + 0.680·6-s − 0.377·7-s + 0.353·8-s − 0.0740·9-s + 0.316·10-s − 0.0274·11-s + 0.481·12-s + 0.149·13-s − 0.267·14-s + 0.430·15-s + 1/4·16-s − 0.727·17-s − 0.0523·18-s + 0.362·19-s + 0.223·20-s − 0.363·21-s − 0.0193·22-s − 0.453·23-s + 0.340·24-s + 1/5·25-s + 0.105·26-s − 1.03·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.762972371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.762972371\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 - 5 T + p^{3} T^{2} \) |
| 11 | \( 1 + T + p^{3} T^{2} \) |
| 13 | \( 1 - 7 T + p^{3} T^{2} \) |
| 17 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 19 | \( 1 - 30 T + p^{3} T^{2} \) |
| 23 | \( 1 + 50 T + p^{3} T^{2} \) |
| 29 | \( 1 - 79 T + p^{3} T^{2} \) |
| 31 | \( 1 + 212 T + p^{3} T^{2} \) |
| 37 | \( 1 + 190 T + p^{3} T^{2} \) |
| 41 | \( 1 + 308 T + p^{3} T^{2} \) |
| 43 | \( 1 - 422 T + p^{3} T^{2} \) |
| 47 | \( 1 - 121 T + p^{3} T^{2} \) |
| 53 | \( 1 - 664 T + p^{3} T^{2} \) |
| 59 | \( 1 - 628 T + p^{3} T^{2} \) |
| 61 | \( 1 + 684 T + p^{3} T^{2} \) |
| 67 | \( 1 - 1056 T + p^{3} T^{2} \) |
| 71 | \( 1 - 744 T + p^{3} T^{2} \) |
| 73 | \( 1 - 726 T + p^{3} T^{2} \) |
| 79 | \( 1 + 407 T + p^{3} T^{2} \) |
| 83 | \( 1 - 644 T + p^{3} T^{2} \) |
| 89 | \( 1 + 880 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1351 T + p^{3} T^{2} \) |
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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
−14.02877161294499100316632927655, −13.44594765697618113745660725958, −12.29312693211253801967469387058, −10.91024816717137091636211064660, −9.554797129906410198586835726130, −8.434841000984463701430404146740, −6.95881400190180065163549875993, −5.53867578225588174848636045224, −3.73453652326893131111598181444, −2.33720109746932864701269806985,
2.33720109746932864701269806985, 3.73453652326893131111598181444, 5.53867578225588174848636045224, 6.95881400190180065163549875993, 8.434841000984463701430404146740, 9.554797129906410198586835726130, 10.91024816717137091636211064660, 12.29312693211253801967469387058, 13.44594765697618113745660725958, 14.02877161294499100316632927655
