L(s) = 1 | + 2-s + 3-s + 4-s − 4·5-s + 6-s − 2·7-s + 8-s + 9-s − 4·10-s − 3·11-s + 12-s − 13-s − 2·14-s − 4·15-s + 16-s − 7·17-s + 18-s − 5·19-s − 4·20-s − 2·21-s − 3·22-s + 4·23-s + 24-s + 11·25-s − 26-s + 27-s − 2·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 1.26·10-s − 0.904·11-s + 0.288·12-s − 0.277·13-s − 0.534·14-s − 1.03·15-s + 1/4·16-s − 1.69·17-s + 0.235·18-s − 1.14·19-s − 0.894·20-s − 0.436·21-s − 0.639·22-s + 0.834·23-s + 0.204·24-s + 11/5·25-s − 0.196·26-s + 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 834 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 834 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 139 | \( 1 + T \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 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
−9.864881560234444862713111158936, −8.601309256469704985679713956813, −8.150475819354056643536211398110, −7.06449682939372982450761368402, −6.60305756728717552187460729760, −4.94851772639154862432578069651, −4.24272235212078727358228108207, −3.36373033086053161208519704185, −2.47183382626775911089007214904, 0,
2.47183382626775911089007214904, 3.36373033086053161208519704185, 4.24272235212078727358228108207, 4.94851772639154862432578069651, 6.60305756728717552187460729760, 7.06449682939372982450761368402, 8.150475819354056643536211398110, 8.601309256469704985679713956813, 9.864881560234444862713111158936