| L(s) = 1 | − 1.79·2-s − 3-s + 1.20·4-s + 1.79·6-s − 7-s + 1.41·8-s + 9-s + 5·11-s − 1.20·12-s + 4.58·13-s + 1.79·14-s − 4.95·16-s + 3·17-s − 1.79·18-s + 3.58·19-s + 21-s − 8.95·22-s + 4·23-s − 1.41·24-s − 8.20·26-s − 27-s − 1.20·28-s + 29-s + 4·31-s + 6.04·32-s − 5·33-s − 5.37·34-s + ⋯ |
| L(s) = 1 | − 1.26·2-s − 0.577·3-s + 0.604·4-s + 0.731·6-s − 0.377·7-s + 0.501·8-s + 0.333·9-s + 1.50·11-s − 0.348·12-s + 1.27·13-s + 0.478·14-s − 1.23·16-s + 0.727·17-s − 0.422·18-s + 0.821·19-s + 0.218·21-s − 1.90·22-s + 0.834·23-s − 0.289·24-s − 1.60·26-s − 0.192·27-s − 0.228·28-s + 0.185·29-s + 0.718·31-s + 1.06·32-s − 0.870·33-s − 0.921·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8845952162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8845952162\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.79T + 2T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 - 4.58T + 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 3.58T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 9.16T + 41T^{2} \) |
| 43 | \( 1 - 9.58T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 0.417T + 53T^{2} \) |
| 59 | \( 1 + 7.58T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 4.16T + 67T^{2} \) |
| 71 | \( 1 + 9.58T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 7.58T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 - 1.41T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−9.162116754132302553154066006700, −8.442482924588290781486228239667, −7.65815231598995003054480680296, −6.71364743734635390796179695737, −6.30162114434693409334273495843, −5.17675596705504956908515818125, −4.13960371082335566514743268559, −3.23214326805480175550137597333, −1.49958259453774684915816459667, −0.881935906972457734636824661786,
0.881935906972457734636824661786, 1.49958259453774684915816459667, 3.23214326805480175550137597333, 4.13960371082335566514743268559, 5.17675596705504956908515818125, 6.30162114434693409334273495843, 6.71364743734635390796179695737, 7.65815231598995003054480680296, 8.442482924588290781486228239667, 9.162116754132302553154066006700
