L(s) = 1 | − 4.83i·2-s − 7.08·3-s − 15.3·4-s + 9.07i·5-s + 34.2i·6-s − 2.15i·7-s + 35.4i·8-s + 23.2·9-s + 43.8·10-s + 11i·11-s + 108.·12-s + (33.3 − 32.9i)13-s − 10.4·14-s − 64.3i·15-s + 48.4·16-s − 49.8·17-s + ⋯ |
L(s) = 1 | − 1.70i·2-s − 1.36·3-s − 1.91·4-s + 0.811i·5-s + 2.32i·6-s − 0.116i·7-s + 1.56i·8-s + 0.860·9-s + 1.38·10-s + 0.301i·11-s + 2.61·12-s + (0.711 − 0.702i)13-s − 0.198·14-s − 1.10i·15-s + 0.756·16-s − 0.711·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.674124 - 0.281967i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.674124 - 0.281967i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - 11iT \) |
| 13 | \( 1 + (-33.3 + 32.9i)T \) |
good | 2 | \( 1 + 4.83iT - 8T^{2} \) |
| 3 | \( 1 + 7.08T + 27T^{2} \) |
| 5 | \( 1 - 9.07iT - 125T^{2} \) |
| 7 | \( 1 + 2.15iT - 343T^{2} \) |
| 17 | \( 1 + 49.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 95.3iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 96.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 115.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 78.6iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 13.4iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 122. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 95.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 361. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 617.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 723. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 549.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 554. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 99.2iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 609. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 586.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 5.23iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 789. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.65e3iT - 9.12e5T^{2} \) |
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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
−12.20570239873070981926679789010, −11.37639150841162724294874445763, −10.59794153758252806960850913305, −10.26439950234285349369385673143, −8.731584265632621585460344070737, −6.91722667846623213097825355049, −5.64119636893547155769752581949, −4.28644485750058674695343799626, −2.89271824540548132667926627147, −1.07854548727815336918077989682,
0.55627688215338095153922169149, 4.52264676341933255034747402166, 5.22486049313722991914378643024, 6.28439756274940341346209702116, 6.97892894247563594943551795692, 8.520975801139652047196573712292, 9.159275783786022739362419583801, 10.85574843323790638190880650972, 11.80154260718933042766805958668, 13.04695865771930109490836850701