L(s) = 1 | + (−1.68 − 1.08i)2-s + (−1.10 − 7.65i)3-s + (1.66 + 3.63i)4-s + (−4.79 − 1.40i)5-s + (−6.42 + 14.0i)6-s + (−11.9 + 13.7i)7-s + (1.13 − 7.91i)8-s + (−31.4 + 9.22i)9-s + (6.54 + 7.55i)10-s + (10.1 − 6.52i)11-s + (26.0 − 16.7i)12-s + (59.7 + 68.9i)13-s + (34.9 − 10.2i)14-s + (−5.50 + 38.2i)15-s + (−10.4 + 12.0i)16-s + (−23.5 + 51.4i)17-s + ⋯ |
L(s) = 1 | + (−0.594 − 0.382i)2-s + (−0.211 − 1.47i)3-s + (0.207 + 0.454i)4-s + (−0.429 − 0.125i)5-s + (−0.436 + 0.956i)6-s + (−0.644 + 0.743i)7-s + (0.0503 − 0.349i)8-s + (−1.16 + 0.341i)9-s + (0.207 + 0.238i)10-s + (0.278 − 0.178i)11-s + (0.625 − 0.402i)12-s + (1.27 + 1.47i)13-s + (0.667 − 0.195i)14-s + (−0.0946 + 0.658i)15-s + (−0.163 + 0.188i)16-s + (−0.335 + 0.734i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0661i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.799828 - 0.0265028i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.799828 - 0.0265028i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.68 + 1.08i)T \) |
| 5 | \( 1 + (4.79 + 1.40i)T \) |
| 23 | \( 1 + (80.7 - 75.1i)T \) |
good | 3 | \( 1 + (1.10 + 7.65i)T + (-25.9 + 7.60i)T^{2} \) |
| 7 | \( 1 + (11.9 - 13.7i)T + (-48.8 - 339. i)T^{2} \) |
| 11 | \( 1 + (-10.1 + 6.52i)T + (552. - 1.21e3i)T^{2} \) |
| 13 | \( 1 + (-59.7 - 68.9i)T + (-312. + 2.17e3i)T^{2} \) |
| 17 | \( 1 + (23.5 - 51.4i)T + (-3.21e3 - 3.71e3i)T^{2} \) |
| 19 | \( 1 + (18.4 + 40.3i)T + (-4.49e3 + 5.18e3i)T^{2} \) |
| 29 | \( 1 + (-109. + 240. i)T + (-1.59e4 - 1.84e4i)T^{2} \) |
| 31 | \( 1 + (20.9 - 145. i)T + (-2.85e4 - 8.39e3i)T^{2} \) |
| 37 | \( 1 + (-184. + 54.2i)T + (4.26e4 - 2.73e4i)T^{2} \) |
| 41 | \( 1 + (-217. - 63.8i)T + (5.79e4 + 3.72e4i)T^{2} \) |
| 43 | \( 1 + (-55.5 - 386. i)T + (-7.62e4 + 2.23e4i)T^{2} \) |
| 47 | \( 1 + 352.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-101. + 117. i)T + (-2.11e4 - 1.47e5i)T^{2} \) |
| 59 | \( 1 + (-51.7 - 59.7i)T + (-2.92e4 + 2.03e5i)T^{2} \) |
| 61 | \( 1 + (-48.2 + 335. i)T + (-2.17e5 - 6.39e4i)T^{2} \) |
| 67 | \( 1 + (-614. - 394. i)T + (1.24e5 + 2.73e5i)T^{2} \) |
| 71 | \( 1 + (-383. - 246. i)T + (1.48e5 + 3.25e5i)T^{2} \) |
| 73 | \( 1 + (-443. - 970. i)T + (-2.54e5 + 2.93e5i)T^{2} \) |
| 79 | \( 1 + (696. + 803. i)T + (-7.01e4 + 4.88e5i)T^{2} \) |
| 83 | \( 1 + (-220. + 64.6i)T + (4.81e5 - 3.09e5i)T^{2} \) |
| 89 | \( 1 + (-182. - 1.26e3i)T + (-6.76e5 + 1.98e5i)T^{2} \) |
| 97 | \( 1 + (640. + 188. i)T + (7.67e5 + 4.93e5i)T^{2} \) |
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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
−11.62499306460675064675961202338, −11.28848938704920799377227201689, −9.607732714448629053772803659787, −8.664208931945159453237069743882, −7.912179629222691012856819837370, −6.57791787352644089379686435524, −6.19494783129150349696328404975, −3.97466368242454381812300615965, −2.34369335453401671818677408199, −1.15053843246339495207088502433,
0.48643645531642948174804051685, 3.29156315935002374169605900378, 4.24000682866299447057282031181, 5.55594464602661490455209568217, 6.68107034794265271999323020834, 7.977279484806482223030576839816, 8.958762796948780163670305916319, 9.995068499869858532352539831134, 10.54795026523285871103579504270, 11.22755215341623714429100314947