| L(s) = 1 | + (10.4 + 18.0i)2-s + (−22.7 + 39.4i)3-s + (−152. + 264. i)4-s + (96.7 + 167. i)5-s − 947.·6-s − 3.69e3·8-s + (58.3 + 101. i)9-s + (−2.01e3 + 3.48e3i)10-s + (−151. + 263. i)11-s + (−6.95e3 − 1.20e4i)12-s + 9.89e3·13-s − 8.80e3·15-s + (−1.89e4 − 3.28e4i)16-s + (6.24e3 − 1.08e4i)17-s + (−1.21e3 + 2.10e3i)18-s + (561. + 973. i)19-s + ⋯ |
| L(s) = 1 | + (0.920 + 1.59i)2-s + (−0.486 + 0.842i)3-s + (−1.19 + 2.06i)4-s + (0.345 + 0.599i)5-s − 1.79·6-s − 2.55·8-s + (0.0266 + 0.0462i)9-s + (−0.636 + 1.10i)10-s + (−0.0344 + 0.0596i)11-s + (−1.16 − 2.01i)12-s + 1.24·13-s − 0.673·15-s + (−1.15 − 2.00i)16-s + (0.308 − 0.533i)17-s + (−0.0490 + 0.0850i)18-s + (0.0187 + 0.0325i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.28983 - 1.93907i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28983 - 1.93907i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (-10.4 - 18.0i)T + (-64 + 110. i)T^{2} \) |
| 3 | \( 1 + (22.7 - 39.4i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-96.7 - 167. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 11 | \( 1 + (151. - 263. i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 - 9.89e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + (-6.24e3 + 1.08e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-561. - 973. i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-2.23e4 - 3.87e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 - 5.12e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + (-1.28e4 + 2.21e4i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-2.10e4 - 3.64e4i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + 7.87e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.29e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-3.13e5 - 5.42e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-2.68e5 + 4.64e5i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-1.06e6 + 1.83e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.35e6 - 2.35e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.88e6 - 3.26e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 - 4.31e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (3.55e5 - 6.14e5i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (1.39e6 + 2.41e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + 5.38e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + (1.09e6 + 1.90e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.16e7T + 8.07e13T^{2} \) |
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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
−15.09297757450467540913896015908, −13.98342874016978574429393864215, −13.17238587341844864009348415862, −11.51468815535617802561516647648, −10.08494903440904316149159757540, −8.485629389939090519629416131811, −7.03248733525600394715894307951, −5.90420232445487789382213237033, −4.84887499452842528784161981685, −3.50435125087861130262064421029,
0.795258230779835285083818683818, 1.71921532447988581587813440944, 3.58498752281438308537703711407, 5.17293440355572766243838462928, 6.37675767293908330337336792403, 8.761101279904336561676841502636, 10.19022729323399734205318667343, 11.32979706846614651518151643776, 12.28380582945381766967970478040, 13.05631451991657641196277641717
