L(s) = 1 | − 4i·7-s + 11-s + 4i·13-s + 6i·17-s − 2·19-s − 4·31-s − 10i·37-s + 4i·43-s − 12i·47-s − 9·49-s + 6i·53-s + 12·59-s − 10·61-s − 4i·67-s − 8i·73-s + ⋯ |
L(s) = 1 | − 1.51i·7-s + 0.301·11-s + 1.10i·13-s + 1.45i·17-s − 0.458·19-s − 0.718·31-s − 1.64i·37-s + 0.609i·43-s − 1.75i·47-s − 1.28·49-s + 0.824i·53-s + 1.56·59-s − 1.28·61-s − 0.488i·67-s − 0.936i·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.220327629\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.220327629\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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
−7.40273294357107614680908933535, −6.77341138342921813035192399707, −6.27075962881103609538249604218, −5.41017717973094151336053205517, −4.36706647135157257728955010571, −4.03531886111023093948189265719, −3.44581037984087268675252139709, −2.09061195760863104520746843774, −1.46414766233713192781759254402, −0.29271962321137785097852004066,
1.00825236056539709944243886946, 2.20821729810326279194073856196, 2.79822483845677109786299469348, 3.47589390702759675798057411835, 4.61212301098970704146288970480, 5.24220687058520745060007070122, 5.74058301610717291777313916830, 6.47429342256373262100918634846, 7.17377824738533838674407658438, 8.029961633466090761468496123663