L(s) = 1 | − 2-s − 3-s − 4-s + 5-s + 6-s − 3·7-s + 3·8-s − 2·9-s − 10-s + 12-s − 5·13-s + 3·14-s − 15-s − 16-s + 3·17-s + 2·18-s − 19-s − 20-s + 3·21-s − 7·23-s − 3·24-s − 4·25-s + 5·26-s + 5·27-s + 3·28-s − 3·29-s + 30-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 + 1.06·8-s − 2/3·9-s − 0.316·10-s + 0.288·12-s − 1.38·13-s + 0.801·14-s − 0.258·15-s − 1/4·16-s + 0.727·17-s + 0.471·18-s − 0.229·19-s − 0.223·20-s + 0.654·21-s − 1.45·23-s − 0.612·24-s − 4/5·25-s + 0.980·26-s + 0.962·27-s + 0.566·28-s − 0.557·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3336119298\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3336119298\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.619159184248389210790268829180, −8.465848922214887729446516878993, −7.85649939029523801473917002587, −6.86535113552977580278725128719, −6.02291781357434847509828575937, −5.35497377421466880095336728668, −4.40336019829144899822949439307, −3.26026786719576192842823709606, −2.08270609090444390409348198124, −0.42854195730248636535135739890,
0.42854195730248636535135739890, 2.08270609090444390409348198124, 3.26026786719576192842823709606, 4.40336019829144899822949439307, 5.35497377421466880095336728668, 6.02291781357434847509828575937, 6.86535113552977580278725128719, 7.85649939029523801473917002587, 8.465848922214887729446516878993, 9.619159184248389210790268829180