L(s) = 1 | + (7.46 − 2.87i)2-s − 15.5i·3-s + (47.4 − 42.9i)4-s + 18.1·5-s + (−44.8 − 116. i)6-s + 321. i·7-s + (230. − 457. i)8-s − 243·9-s + (135. − 52.1i)10-s + 2.01e3i·11-s + (−669. − 739. i)12-s − 2.84e3·13-s + (924. + 2.39e3i)14-s − 282. i·15-s + (405. − 4.07e3i)16-s + 6.86e3·17-s + ⋯ |
L(s) = 1 | + (0.933 − 0.359i)2-s − 0.577i·3-s + (0.741 − 0.671i)4-s + 0.144·5-s + (−0.207 − 0.538i)6-s + 0.937i·7-s + (0.450 − 0.892i)8-s − 0.333·9-s + (0.135 − 0.0521i)10-s + 1.51i·11-s + (−0.387 − 0.427i)12-s − 1.29·13-s + (0.337 + 0.874i)14-s − 0.0836i·15-s + (0.0990 − 0.995i)16-s + 1.39·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.741i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.671 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.96687 - 0.872492i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.96687 - 0.872492i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-7.46 + 2.87i)T \) |
| 3 | \( 1 + 15.5iT \) |
good | 5 | \( 1 - 18.1T + 1.56e4T^{2} \) |
| 7 | \( 1 - 321. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 2.01e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 2.84e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 6.86e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 9.83e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 4.62e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 1.77e3T + 5.94e8T^{2} \) |
| 31 | \( 1 - 1.62e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 4.22e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 1.46e3T + 4.75e9T^{2} \) |
| 43 | \( 1 + 4.12e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 5.60e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 6.51e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + 2.41e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 3.04e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 8.37e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 4.98e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 4.95e4T + 1.51e11T^{2} \) |
| 79 | \( 1 - 8.96e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 6.68e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 2.19e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 1.76e5T + 8.32e11T^{2} \) |
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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
−18.99023373347988796771123569303, −17.46304879474166458829957217286, −15.44240921408206359212344360237, −14.38060844447914146478589727071, −12.67880010916371757062566574082, −11.92822936485764427040221210233, −9.811099566533262993915216706015, −7.13911800758783397918168428277, −5.18725548164028441227712165290, −2.31456411752238870578690078242,
3.63164797324934466661251804311, 5.63815772342515592872505611310, 7.77929066554938547993152078376, 10.26685767160733833444495179792, 11.89946272249858519136939563429, 13.70598050959148834767969199254, 14.63719107779964660401951032110, 16.34679832915544498340652901556, 17.01473424127231338326069167278, 19.37856613498623495773329321689