L(s) = 1 | + 2-s + 3-s + 4-s + 2·5-s + 6-s + 8-s + 9-s + 2·10-s + 2·11-s + 12-s + 6·13-s + 2·15-s + 16-s + 4·17-s + 18-s − 19-s + 2·20-s + 2·22-s − 4·23-s + 24-s − 25-s + 6·26-s + 27-s − 2·29-s + 2·30-s + 6·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.603·11-s + 0.288·12-s + 1.66·13-s + 0.516·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.229·19-s + 0.447·20-s + 0.426·22-s − 0.834·23-s + 0.204·24-s − 1/5·25-s + 1.17·26-s + 0.192·27-s − 0.371·29-s + 0.365·30-s + 1.07·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.565787989\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.565787989\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.269672212103568791099153634346, −7.32758376882479251122251264425, −6.51048390401872982405003956746, −5.96425538797600495431959366808, −5.41546051952459436983822074038, −4.28344042944508400009897179721, −3.69849192810968923563655438589, −2.95879653157062806533221586269, −1.89380219938834387298445463361, −1.26563214064963738536093444692,
1.26563214064963738536093444692, 1.89380219938834387298445463361, 2.95879653157062806533221586269, 3.69849192810968923563655438589, 4.28344042944508400009897179721, 5.41546051952459436983822074038, 5.96425538797600495431959366808, 6.51048390401872982405003956746, 7.32758376882479251122251264425, 8.269672212103568791099153634346