| L(s) = 1 | + (−22.6 + 39.1i)2-s + (−702. + 405. i)3-s + (−1.02e3 − 1.77e3i)4-s + (1.83e4 + 1.05e4i)5-s − 3.67e4i·6-s + 9.26e4·8-s + (6.33e4 − 1.09e5i)9-s + (−8.28e5 + 4.78e5i)10-s + (6.51e5 + 1.12e6i)11-s + (1.43e6 + 8.30e5i)12-s − 7.09e6i·13-s − 1.71e7·15-s + (−2.09e6 + 3.63e6i)16-s + (2.79e7 − 1.61e7i)17-s + (2.86e6 + 4.96e6i)18-s + (2.32e7 + 1.34e7i)19-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.963 + 0.556i)3-s + (−0.249 − 0.433i)4-s + (1.17 + 0.676i)5-s − 0.786i·6-s + 0.353·8-s + (0.119 − 0.206i)9-s + (−0.828 + 0.478i)10-s + (0.367 + 0.637i)11-s + (0.481 + 0.278i)12-s − 1.47i·13-s − 1.50·15-s + (−0.125 + 0.216i)16-s + (1.15 − 0.667i)17-s + (0.0843 + 0.146i)18-s + (0.493 + 0.285i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.09136420258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09136420258\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (22.6 - 39.1i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (702. - 405. i)T + (2.65e5 - 4.60e5i)T^{2} \) |
| 5 | \( 1 + (-1.83e4 - 1.05e4i)T + (1.22e8 + 2.11e8i)T^{2} \) |
| 11 | \( 1 + (-6.51e5 - 1.12e6i)T + (-1.56e12 + 2.71e12i)T^{2} \) |
| 13 | \( 1 + 7.09e6iT - 2.32e13T^{2} \) |
| 17 | \( 1 + (-2.79e7 + 1.61e7i)T + (2.91e14 - 5.04e14i)T^{2} \) |
| 19 | \( 1 + (-2.32e7 - 1.34e7i)T + (1.10e15 + 1.91e15i)T^{2} \) |
| 23 | \( 1 + (-6.22e7 + 1.07e8i)T + (-1.09e16 - 1.89e16i)T^{2} \) |
| 29 | \( 1 + 7.52e8T + 3.53e17T^{2} \) |
| 31 | \( 1 + (9.46e8 - 5.46e8i)T + (3.93e17 - 6.82e17i)T^{2} \) |
| 37 | \( 1 + (1.93e9 - 3.35e9i)T + (-3.29e18 - 5.70e18i)T^{2} \) |
| 41 | \( 1 - 2.55e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 + 4.36e9T + 3.99e19T^{2} \) |
| 47 | \( 1 + (8.04e9 + 4.64e9i)T + (5.80e19 + 1.00e20i)T^{2} \) |
| 53 | \( 1 + (1.09e8 + 1.89e8i)T + (-2.45e20 + 4.25e20i)T^{2} \) |
| 59 | \( 1 + (5.45e10 - 3.14e10i)T + (8.89e20 - 1.54e21i)T^{2} \) |
| 61 | \( 1 + (4.19e10 + 2.42e10i)T + (1.32e21 + 2.29e21i)T^{2} \) |
| 67 | \( 1 + (5.49e10 + 9.51e10i)T + (-4.09e21 + 7.08e21i)T^{2} \) |
| 71 | \( 1 + 7.43e10T + 1.64e22T^{2} \) |
| 73 | \( 1 + (-2.26e11 + 1.30e11i)T + (1.14e22 - 1.98e22i)T^{2} \) |
| 79 | \( 1 + (1.20e11 - 2.08e11i)T + (-2.95e22 - 5.11e22i)T^{2} \) |
| 83 | \( 1 - 2.09e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 + (1.79e10 + 1.03e10i)T + (1.23e23 + 2.13e23i)T^{2} \) |
| 97 | \( 1 + 4.74e11iT - 6.93e23T^{2} \) |
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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
−10.85227001334459187116730267239, −10.17097005290053055649027068725, −9.489248459568739461210006013187, −7.82823232429010077458328171282, −6.63227861813163668300913048413, −5.61526183265695000449153590096, −5.05147475346625113297027956709, −3.10715045958268221593798748514, −1.48115504643525668305127706870, −0.02697521799276636174057848101,
1.26677365773772170822872592417, 1.76883015748504906766796428904, 3.63589386083594710401047656002, 5.30485146709740190197764648423, 6.03637305498345326094443743790, 7.31261377287673909199120196403, 8.997236593405862883108696135588, 9.525724307181341177632545943071, 10.94789110540691523635443910635, 11.73881509987598206820980251580
