L(s) = 1 | + (−2.80 + 0.325i)2-s − 3i·3-s + (7.78 − 1.83i)4-s − 18.5i·5-s + (0.977 + 8.42i)6-s + 9.32·7-s + (−21.2 + 7.68i)8-s − 9·9-s + (6.04 + 52.0i)10-s + 39.7i·11-s + (−5.49 − 23.3i)12-s − 32.9i·13-s + (−26.2 + 3.04i)14-s − 55.6·15-s + (57.2 − 28.5i)16-s + 90.5·17-s + ⋯ |
L(s) = 1 | + (−0.993 + 0.115i)2-s − 0.577i·3-s + (0.973 − 0.228i)4-s − 1.65i·5-s + (0.0665 + 0.573i)6-s + 0.503·7-s + (−0.940 + 0.339i)8-s − 0.333·9-s + (0.191 + 1.64i)10-s + 1.08i·11-s + (−0.132 − 0.562i)12-s − 0.703i·13-s + (−0.500 + 0.0580i)14-s − 0.957·15-s + (0.895 − 0.445i)16-s + 1.29·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.339 + 0.940i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.339 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.636769 - 0.447083i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.636769 - 0.447083i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.80 - 0.325i)T \) |
| 3 | \( 1 + 3iT \) |
good | 5 | \( 1 + 18.5iT - 125T^{2} \) |
| 7 | \( 1 - 9.32T + 343T^{2} \) |
| 11 | \( 1 - 39.7iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 32.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 90.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 72.5iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 45.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 143. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 90.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 1.77iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 195.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 407. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 278.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 241. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 149. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 508. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 950. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 803.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 449.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 157.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 175. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 127.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 158.T + 9.12e5T^{2} \) |
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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
−17.14476168075069414521358555668, −16.19794799803469453426678271317, −14.71019484089938583453067523721, −12.73978053989557095084042727127, −11.96299671768611326057438051463, −10.01859238145999464636812286335, −8.605645948350586607916413881782, −7.59343552523933760602606176166, −5.39861801376524776329793114427, −1.32506279373975630882762425965,
3.05990531721135832667930574220, 6.34884103545844015933093148626, 7.895849629935423211689236915711, 9.605669855423760745739101161610, 10.87213452841818450821183791433, 11.51053377909254462772901100762, 14.13831872233514156441667605230, 15.11585186432710096099572868342, 16.36636102613912267939657393564, 17.61624454576484316199602628817