| L(s) = 1 | − 57.1i·2-s + 18.6i·3-s − 2.23e3·4-s + 2.20e3i·5-s + 1.06e3·6-s + 1.81e3i·7-s + 6.92e4i·8-s + 5.87e4·9-s + 1.25e5·10-s + 8.48e4·11-s − 4.17e4i·12-s − 1.96e5·13-s + 1.03e5·14-s − 4.10e4·15-s + 1.66e6·16-s + 5.62e5·17-s + ⋯ |
| L(s) = 1 | − 1.78i·2-s + 0.0767i·3-s − 2.18·4-s + 0.704i·5-s + 0.136·6-s + 0.108i·7-s + 2.11i·8-s + 0.994·9-s + 1.25·10-s + 0.527·11-s − 0.167i·12-s − 0.529·13-s + 0.193·14-s − 0.0540·15-s + 1.58·16-s + 0.395·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.831844 - 1.60112i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.831844 - 1.60112i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-8.45e7 + 1.20e8i)T \) |
| good | 2 | \( 1 + 57.1iT - 1.02e3T^{2} \) |
| 3 | \( 1 - 18.6iT - 5.90e4T^{2} \) |
| 5 | \( 1 - 2.20e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 - 1.81e3iT - 2.82e8T^{2} \) |
| 11 | \( 1 - 8.48e4T + 2.59e10T^{2} \) |
| 13 | \( 1 + 1.96e5T + 1.37e11T^{2} \) |
| 17 | \( 1 - 5.62e5T + 2.01e12T^{2} \) |
| 19 | \( 1 + 2.47e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 - 4.25e6T + 4.14e13T^{2} \) |
| 29 | \( 1 - 7.15e3iT - 4.20e14T^{2} \) |
| 31 | \( 1 - 9.25e6T + 8.19e14T^{2} \) |
| 37 | \( 1 + 2.64e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 + 1.16e8T + 1.34e16T^{2} \) |
| 47 | \( 1 - 3.14e7T + 5.25e16T^{2} \) |
| 53 | \( 1 - 2.99e8T + 1.74e17T^{2} \) |
| 59 | \( 1 - 6.98e6T + 5.11e17T^{2} \) |
| 61 | \( 1 + 1.07e9iT - 7.13e17T^{2} \) |
| 67 | \( 1 - 1.30e9T + 1.82e18T^{2} \) |
| 71 | \( 1 + 1.39e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 7.73e8iT - 4.29e18T^{2} \) |
| 79 | \( 1 + 2.98e8T + 9.46e18T^{2} \) |
| 83 | \( 1 - 3.52e9T + 1.55e19T^{2} \) |
| 89 | \( 1 + 8.07e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + 7.54e9T + 7.37e19T^{2} \) |
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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
−12.98142955218668445343653316147, −12.01498993987638435326431327780, −10.91921611305114033181210734231, −10.04452658865412375406945534338, −9.006779972825428091439186952849, −7.04472343252124655346790266707, −4.79256555740882594312634463128, −3.48783003067293310711487824074, −2.24137700929785494338427144649, −0.810480869031902552764870608221,
1.02630456561437440512285416711, 4.16560349468180984601118606912, 5.26654786895526743958189439645, 6.66176911704819389384163788389, 7.69041012199834427076925963833, 8.858000891300140611521213068469, 9.985493806169127579563382586479, 12.29075470998073675193977450341, 13.30668820364947235507615510799, 14.45469091629507871036956049648
