L(s) = 1 | + 2-s − 2·3-s + 4-s − 5-s − 2·6-s + 7-s + 8-s + 9-s − 10-s − 6·11-s − 2·12-s − 4·13-s + 14-s + 2·15-s + 16-s + 6·17-s + 18-s + 8·19-s − 20-s − 2·21-s − 6·22-s + 23-s − 2·24-s + 25-s − 4·26-s + 4·27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.80·11-s − 0.577·12-s − 1.10·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 1.83·19-s − 0.223·20-s − 0.436·21-s − 1.27·22-s + 0.208·23-s − 0.408·24-s + 1/5·25-s − 0.784·26-s + 0.769·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.417364992\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.417364992\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.804138627097304869465745836991, −8.285746504554624384872788431081, −7.55884584235795011657377627575, −7.05520108669387958265164424169, −5.72447800456945246360797480173, −5.21022855726840300708586043221, −4.89123141179020227048767856254, −3.42366599428342128168538377727, −2.55080738571920609620281269689, −0.78351974673693313575593831890,
0.78351974673693313575593831890, 2.55080738571920609620281269689, 3.42366599428342128168538377727, 4.89123141179020227048767856254, 5.21022855726840300708586043221, 5.72447800456945246360797480173, 7.05520108669387958265164424169, 7.55884584235795011657377627575, 8.285746504554624384872788431081, 9.804138627097304869465745836991