L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 7-s + 8-s + 9-s + 10-s + 6·11-s − 12-s − 4·13-s + 14-s − 15-s + 16-s − 7·17-s + 18-s − 3·19-s + 20-s − 21-s + 6·22-s + 4·23-s − 24-s − 4·25-s − 4·26-s − 27-s + 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 + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s − 0.288·12-s − 1.10·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.69·17-s + 0.235·18-s − 0.688·19-s + 0.223·20-s − 0.218·21-s + 1.27·22-s + 0.834·23-s − 0.204·24-s − 4/5·25-s − 0.784·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.584700771\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.584700771\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 29 | \( 1 + T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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
−12.69382741216715398649974147513, −11.73639879382114088109004338099, −11.10440562935940046336115880040, −9.836098981125502929268702372114, −8.771228911258239129730405843616, −7.03592383217191795133937185850, −6.37937256584181034035834610985, −5.04777884675962662359834544429, −4.05020215715234580755213266403, −1.99193874858937465855035057059,
1.99193874858937465855035057059, 4.05020215715234580755213266403, 5.04777884675962662359834544429, 6.37937256584181034035834610985, 7.03592383217191795133937185850, 8.771228911258239129730405843616, 9.836098981125502929268702372114, 11.10440562935940046336115880040, 11.73639879382114088109004338099, 12.69382741216715398649974147513