L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 3·7-s − 8-s + 9-s + 10-s + 11-s + 12-s + 13-s + 3·14-s − 15-s + 16-s − 17-s − 18-s − 3·19-s − 20-s − 3·21-s − 22-s − 23-s − 24-s + 25-s − 26-s + 27-s − 3·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.277·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.688·19-s − 0.223·20-s − 0.654·21-s − 0.213·22-s − 0.208·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
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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
−7.913531864345149661497224499812, −7.15651321242440418914649345100, −6.47535349767769256235190525128, −6.01471798832912856503188497824, −4.67784908219683923384334566335, −3.90987475651168139228794899714, −3.11202925640995898839969368780, −2.43586097740355171487913877484, −1.22056516522996576839607585878, 0,
1.22056516522996576839607585878, 2.43586097740355171487913877484, 3.11202925640995898839969368780, 3.90987475651168139228794899714, 4.67784908219683923384334566335, 6.01471798832912856503188497824, 6.47535349767769256235190525128, 7.15651321242440418914649345100, 7.913531864345149661497224499812