| L(s) = 1 | + (−4.29 + 3.68i)2-s + (−1.79 + 3.10i)3-s + (4.84 − 31.6i)4-s + (−10.1 + 17.5i)5-s + (−3.74 − 19.9i)6-s − 149. i·7-s + (95.7 + 153. i)8-s + (115. + 199. i)9-s + (−21.1 − 112. i)10-s + 123. i·11-s + (89.4 + 71.7i)12-s + (412. − 238. i)13-s + (552. + 643. i)14-s + (−36.2 − 62.7i)15-s + (−976. − 306. i)16-s + (−809. + 1.40e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.758 + 0.651i)2-s + (−0.114 + 0.199i)3-s + (0.151 − 0.988i)4-s + (−0.180 + 0.313i)5-s + (−0.0424 − 0.225i)6-s − 1.15i·7-s + (0.528 + 0.848i)8-s + (0.473 + 0.820i)9-s + (−0.0668 − 0.355i)10-s + 0.308i·11-s + (0.179 + 0.143i)12-s + (0.677 − 0.390i)13-s + (0.753 + 0.877i)14-s + (−0.0415 − 0.0720i)15-s + (−0.954 − 0.299i)16-s + (−0.679 + 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.583 - 0.812i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.583 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.401524 + 0.782596i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.401524 + 0.782596i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.29 - 3.68i)T \) |
| 19 | \( 1 + (-1.55e3 + 240. i)T \) |
| good | 3 | \( 1 + (1.79 - 3.10i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (10.1 - 17.5i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + 149. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 123. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-412. + 238. i)T + (1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + (809. - 1.40e3i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 23 | \( 1 + (3.75e3 - 2.16e3i)T + (3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (6.56e3 - 3.79e3i)T + (1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + 3.22e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.80e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + (-1.58e4 - 9.17e3i)T + (5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-9.28e3 - 5.35e3i)T + (7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-2.58e4 + 1.48e4i)T + (1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (2.42e4 - 1.40e4i)T + (2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (6.49e3 - 1.12e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (609. + 1.05e3i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.05e4 - 1.82e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-5.15e3 + 8.92e3i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-1.23e4 + 2.13e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (3.20e4 - 5.55e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 6.23e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 + (-3.75e4 + 2.17e4i)T + (2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + (-1.95e3 - 1.12e3i)T + (4.29e9 + 7.43e9i)T^{2} \) |
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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
−14.03380119933086730269121753042, −13.12954144419033502743278371445, −11.07898083361692473004060793612, −10.58321060251408538142258793652, −9.446160846498229037345345672412, −7.86807897150981632706310556759, −7.20432957918881685337364136986, −5.68947950661125520943262410391, −4.07475035659360929573041222635, −1.46649084400630640894015882537,
0.52913557707429353705062012070, 2.28587828910719534709734413420, 4.00412795528928830203664043095, 6.04740534267071405167636597659, 7.53638411881007098570261424545, 8.906609447230767378144409263617, 9.470580358778422001558959589242, 11.08976773957109733562789747048, 12.01619660537475897808799169584, 12.64776608616913703926882516550
