| L(s) = 1 | + 44.1i·2-s + 389. i·3-s − 927.·4-s − 5.23e3i·5-s − 1.71e4·6-s − 2.18e4i·7-s + 4.27e3i·8-s − 9.24e4·9-s + 2.31e5·10-s + 1.62e4·11-s − 3.60e5i·12-s + 3.26e5·13-s + 9.67e5·14-s + 2.03e6·15-s − 1.13e6·16-s + 2.87e5·17-s + ⋯ |
| L(s) = 1 | + 1.38i·2-s + 1.60i·3-s − 0.905·4-s − 1.67i·5-s − 2.21·6-s − 1.30i·7-s + 0.130i·8-s − 1.56·9-s + 2.31·10-s + 0.100·11-s − 1.45i·12-s + 0.878·13-s + 1.79·14-s + 2.68·15-s − 1.08·16-s + 0.202·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.51481 + 0.928471i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51481 + 0.928471i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (6.67e7 - 1.30e8i)T \) |
| good | 2 | \( 1 - 44.1iT - 1.02e3T^{2} \) |
| 3 | \( 1 - 389. iT - 5.90e4T^{2} \) |
| 5 | \( 1 + 5.23e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 + 2.18e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 - 1.62e4T + 2.59e10T^{2} \) |
| 13 | \( 1 - 3.26e5T + 1.37e11T^{2} \) |
| 17 | \( 1 - 2.87e5T + 2.01e12T^{2} \) |
| 19 | \( 1 + 1.39e5iT - 6.13e12T^{2} \) |
| 23 | \( 1 - 1.90e6T + 4.14e13T^{2} \) |
| 29 | \( 1 + 3.83e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 - 5.40e7T + 8.19e14T^{2} \) |
| 37 | \( 1 + 8.73e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 - 8.87e7T + 1.34e16T^{2} \) |
| 47 | \( 1 - 1.99e8T + 5.25e16T^{2} \) |
| 53 | \( 1 + 3.58e8T + 1.74e17T^{2} \) |
| 59 | \( 1 - 9.43e8T + 5.11e17T^{2} \) |
| 61 | \( 1 + 1.50e9iT - 7.13e17T^{2} \) |
| 67 | \( 1 + 1.23e9T + 1.82e18T^{2} \) |
| 71 | \( 1 - 1.85e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 1.84e9iT - 4.29e18T^{2} \) |
| 79 | \( 1 + 8.05e8T + 9.46e18T^{2} \) |
| 83 | \( 1 + 6.35e9T + 1.55e19T^{2} \) |
| 89 | \( 1 + 3.36e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 - 1.34e10T + 7.37e19T^{2} \) |
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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
−14.24816840604942010437551710449, −13.28774051470292506014033567968, −11.35150596463974497275532433260, −9.933304165700387732532356963332, −8.872907490498346369175887011015, −7.903653247555090333479551558656, −5.97474501469927418737993152082, −4.72657107756043359080177018274, −4.09919687339472144258736173361, −0.64804307152826066463629428214,
1.26567612365102684636483348264, 2.43759929736859434154833366627, 3.13105712873850286055980090951, 6.12847907638449826868900934980, 7.03787326186116070731715207527, 8.653262354523012920053139217189, 10.36909510262075156652481699378, 11.48568374548139022804209043591, 12.09603083892420887519112957465, 13.21181485920502800513137858279
