| L(s) = 1 | + (11.3 − 7.29i)2-s + (2.21 − 15.4i)3-s + (48.9 − 107. i)4-s + (−38.1 − 129. i)5-s + (−87.3 − 191. i)6-s + (−288. + 249. i)7-s + (−103. − 720. i)8-s + (−233. − 68.4i)9-s + (−1.38e3 − 1.19e3i)10-s + (772. − 1.20e3i)11-s + (−1.54e3 − 993. i)12-s + (−726. + 838. i)13-s + (−1.44e3 + 4.93e3i)14-s + (−2.08e3 + 300. i)15-s + (−1.48e3 − 1.71e3i)16-s + (2.59e3 − 1.18e3i)17-s + ⋯ |
| L(s) = 1 | + (1.41 − 0.911i)2-s + (0.0821 − 0.571i)3-s + (0.765 − 1.67i)4-s + (−0.305 − 1.03i)5-s + (−0.404 − 0.885i)6-s + (−0.839 + 0.727i)7-s + (−0.202 − 1.40i)8-s + (−0.319 − 0.0939i)9-s + (−1.38 − 1.19i)10-s + (0.580 − 0.903i)11-s + (−0.894 − 0.575i)12-s + (−0.330 + 0.381i)13-s + (−0.527 + 1.79i)14-s + (−0.619 + 0.0890i)15-s + (−0.362 − 0.417i)16-s + (0.528 − 0.241i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.136i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.990 + 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.228085 - 3.32939i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.228085 - 3.32939i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.21 + 15.4i)T \) |
| 23 | \( 1 + (-869. + 1.21e4i)T \) |
| good | 2 | \( 1 + (-11.3 + 7.29i)T + (26.5 - 58.2i)T^{2} \) |
| 5 | \( 1 + (38.1 + 129. i)T + (-1.31e4 + 8.44e3i)T^{2} \) |
| 7 | \( 1 + (288. - 249. i)T + (1.67e4 - 1.16e5i)T^{2} \) |
| 11 | \( 1 + (-772. + 1.20e3i)T + (-7.35e5 - 1.61e6i)T^{2} \) |
| 13 | \( 1 + (726. - 838. i)T + (-6.86e5 - 4.77e6i)T^{2} \) |
| 17 | \( 1 + (-2.59e3 + 1.18e3i)T + (1.58e7 - 1.82e7i)T^{2} \) |
| 19 | \( 1 + (-2.68e3 - 1.22e3i)T + (3.08e7 + 3.55e7i)T^{2} \) |
| 29 | \( 1 + (1.66e4 + 3.64e4i)T + (-3.89e8 + 4.49e8i)T^{2} \) |
| 31 | \( 1 + (1.64e3 + 1.14e4i)T + (-8.51e8 + 2.50e8i)T^{2} \) |
| 37 | \( 1 + (2.17e4 - 7.40e4i)T + (-2.15e9 - 1.38e9i)T^{2} \) |
| 41 | \( 1 + (-9.63e4 + 2.82e4i)T + (3.99e9 - 2.56e9i)T^{2} \) |
| 43 | \( 1 + (5.78e4 + 8.32e3i)T + (6.06e9 + 1.78e9i)T^{2} \) |
| 47 | \( 1 - 1.45e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + (-7.95e4 + 6.89e4i)T + (3.15e9 - 2.19e10i)T^{2} \) |
| 59 | \( 1 + (-1.09e5 + 1.26e5i)T + (-6.00e9 - 4.17e10i)T^{2} \) |
| 61 | \( 1 + (-3.15e5 + 4.53e4i)T + (4.94e10 - 1.45e10i)T^{2} \) |
| 67 | \( 1 + (-2.22e5 - 3.46e5i)T + (-3.75e10 + 8.22e10i)T^{2} \) |
| 71 | \( 1 + (2.27e5 - 1.45e5i)T + (5.32e10 - 1.16e11i)T^{2} \) |
| 73 | \( 1 + (-1.31e5 + 2.87e5i)T + (-9.91e10 - 1.14e11i)T^{2} \) |
| 79 | \( 1 + (-4.02e5 - 3.48e5i)T + (3.45e10 + 2.40e11i)T^{2} \) |
| 83 | \( 1 + (1.79e5 - 6.10e5i)T + (-2.75e11 - 1.76e11i)T^{2} \) |
| 89 | \( 1 + (-1.92e5 - 2.76e4i)T + (4.76e11 + 1.40e11i)T^{2} \) |
| 97 | \( 1 + (1.45e5 + 4.94e5i)T + (-7.00e11 + 4.50e11i)T^{2} \) |
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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
−12.84644232198575203628602164032, −12.12416142950321970929029011021, −11.46594849840650693106415490633, −9.692669626072717171718435796705, −8.413861754329787543955848707177, −6.38002802503209349021771786775, −5.32917716071295254554248105328, −3.87437001091651229593940289406, −2.51566904274349186851137953051, −0.841529832410992901962671770673,
3.18099717600556908734194393529, 3.94266385009092449579174920830, 5.43752532585915489551010496322, 6.87614943734738455364275021737, 7.42337257050202420853776259554, 9.607225232584698595143758241391, 10.80219401155499544675947515913, 12.19473196263171883207508674694, 13.21887666835277153935542401533, 14.42550582708705959284433512856
