L(s) = 1 | + 2-s − 3-s − 4-s − 5-s − 6-s − 2·7-s − 3·8-s + 9-s − 10-s − 6·11-s + 12-s − 2·14-s + 15-s − 16-s − 6·17-s + 18-s + 20-s + 2·21-s − 6·22-s − 8·23-s + 3·24-s + 25-s − 27-s + 2·28-s − 4·29-s + 30-s + 5·32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.755·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s − 1.80·11-s + 0.288·12-s − 0.534·14-s + 0.258·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.223·20-s + 0.436·21-s − 1.27·22-s − 1.66·23-s + 0.612·24-s + 1/5·25-s − 0.192·27-s + 0.377·28-s − 0.742·29-s + 0.182·30-s + 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−7.42195896981757685932141018119, −6.45754305631640026627862079587, −5.96416216153721976810548337042, −5.13514216792987632614840809373, −4.62770227716381985665512695248, −3.81467929255751360490045430008, −3.05985687846521158771439909949, −2.09668804479314175714108769749, 0, 0,
2.09668804479314175714108769749, 3.05985687846521158771439909949, 3.81467929255751360490045430008, 4.62770227716381985665512695248, 5.13514216792987632614840809373, 5.96416216153721976810548337042, 6.45754305631640026627862079587, 7.42195896981757685932141018119