L(s) = 1 | + (2.82 − 2.82i)2-s − 16.0i·4-s + (158. + 158. i)7-s + (−45.2 − 45.2i)8-s + 147. i·11-s + (516. − 516. i)13-s + 897.·14-s − 256.·16-s + (−1.09e3 + 1.09e3i)17-s − 1.17e3i·19-s + (415. + 415. i)22-s + (3.13e3 + 3.13e3i)23-s − 2.92e3i·26-s + (2.53e3 − 2.53e3i)28-s − 1.86e3·29-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (1.22 + 1.22i)7-s + (−0.250 − 0.250i)8-s + 0.366i·11-s + (0.847 − 0.847i)13-s + 1.22·14-s − 0.250·16-s + (−0.917 + 0.917i)17-s − 0.747i·19-s + (0.183 + 0.183i)22-s + (1.23 + 1.23i)23-s − 0.847i·26-s + (0.611 − 0.611i)28-s − 0.411·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.465i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.884 - 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(3.152804838\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.152804838\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.82 + 2.82i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-158. - 158. i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 - 147. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-516. + 516. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (1.09e3 - 1.09e3i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 + 1.17e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (-3.13e3 - 3.13e3i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 + 1.86e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.28e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-1.68e3 - 1.68e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 - 9.09e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (1.37e4 - 1.37e4i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (-887. + 887. i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (-1.63e4 - 1.63e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 - 957.T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.93e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (3.57e4 + 3.57e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 - 2.59e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-5.28e4 + 5.28e4i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 + 1.63e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + (-6.95e4 - 6.95e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 - 1.33e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + (8.35e3 + 8.35e3i)T + 8.58e9iT^{2} \) |
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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
−10.78002715778242702425487768071, −9.375575485345359088044328158119, −8.669765197226058919846868133651, −7.74963596935502790461046713968, −6.35532476897146512264542218042, −5.39583743869675743224615077228, −4.70396863384038619892003666494, −3.36050094654247099037978006926, −2.19338459952226065204815971342, −1.25731669275470770678071483568,
0.66019374126202243136758812408, 1.98843174937285472882003415445, 3.65459086260920463831922560594, 4.44471015112227841581886019703, 5.31997600605902836498862808974, 6.67517947194489969161739447937, 7.25175673515586367307369092431, 8.332405048372863314693438058268, 9.006611126892477786912618483034, 10.50686523560639852612474919392