| L(s) = 1 | + (−108 + 187. i)2-s + (−1.67e3 − 2.89e3i)3-s + (−6.94e3 − 1.20e4i)4-s + (2.60e4 − 4.51e4i)5-s + 7.23e5·6-s − 4.07e6·8-s + (1.56e6 − 2.71e6i)9-s + (5.62e6 + 9.74e6i)10-s + (−1.02e7 − 1.78e7i)11-s + (−2.32e7 + 4.02e7i)12-s + 1.90e8·13-s − 1.74e8·15-s + (6.67e8 − 1.15e9i)16-s + (8.23e8 + 1.42e9i)17-s + (3.39e8 + 5.87e8i)18-s + (7.81e8 − 1.35e9i)19-s + ⋯ |
| L(s) = 1 | + (−0.596 + 1.03i)2-s + (−0.441 − 0.765i)3-s + (−0.211 − 0.367i)4-s + (0.149 − 0.258i)5-s + 1.05·6-s − 0.687·8-s + (0.109 − 0.189i)9-s + (0.177 + 0.308i)10-s + (−0.159 − 0.275i)11-s + (−0.187 + 0.324i)12-s + 0.840·13-s − 0.263·15-s + (0.622 − 1.07i)16-s + (0.486 + 0.842i)17-s + (0.130 + 0.226i)18-s + (0.200 − 0.347i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(0.9958817796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9958817796\) |
| \(L(\frac{17}{2})\) |
|
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 | \( 1 + (108 - 187. i)T + (-1.63e4 - 2.83e4i)T^{2} \) |
| 3 | \( 1 + (1.67e3 + 2.89e3i)T + (-7.17e6 + 1.24e7i)T^{2} \) |
| 5 | \( 1 + (-2.60e4 + 4.51e4i)T + (-1.52e10 - 2.64e10i)T^{2} \) |
| 11 | \( 1 + (1.02e7 + 1.78e7i)T + (-2.08e15 + 3.61e15i)T^{2} \) |
| 13 | \( 1 - 1.90e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + (-8.23e8 - 1.42e9i)T + (-1.43e18 + 2.47e18i)T^{2} \) |
| 19 | \( 1 + (-7.81e8 + 1.35e9i)T + (-7.59e18 - 1.31e19i)T^{2} \) |
| 23 | \( 1 + (4.72e9 - 8.18e9i)T + (-1.33e20 - 2.30e20i)T^{2} \) |
| 29 | \( 1 + 3.69e10T + 8.62e21T^{2} \) |
| 31 | \( 1 + (-3.57e10 - 6.19e10i)T + (-1.17e22 + 2.03e22i)T^{2} \) |
| 37 | \( 1 + (-5.16e11 + 8.95e11i)T + (-1.66e23 - 2.88e23i)T^{2} \) |
| 41 | \( 1 + 1.64e12T + 1.55e24T^{2} \) |
| 43 | \( 1 + 4.92e11T + 3.17e24T^{2} \) |
| 47 | \( 1 + (1.70e12 - 2.95e12i)T + (-6.03e24 - 1.04e25i)T^{2} \) |
| 53 | \( 1 + (3.39e12 + 5.88e12i)T + (-3.65e25 + 6.33e25i)T^{2} \) |
| 59 | \( 1 + (-4.92e12 - 8.53e12i)T + (-1.82e26 + 3.16e26i)T^{2} \) |
| 61 | \( 1 + (-2.46e12 + 4.27e12i)T + (-3.01e26 - 5.21e26i)T^{2} \) |
| 67 | \( 1 + (-1.44e13 - 2.49e13i)T + (-1.23e27 + 2.13e27i)T^{2} \) |
| 71 | \( 1 - 1.25e14T + 5.87e27T^{2} \) |
| 73 | \( 1 + (4.10e13 + 7.11e13i)T + (-4.45e27 + 7.71e27i)T^{2} \) |
| 79 | \( 1 + (-1.27e13 + 2.20e13i)T + (-1.45e28 - 2.52e28i)T^{2} \) |
| 83 | \( 1 - 2.81e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + (-3.57e14 + 6.19e14i)T + (-8.70e28 - 1.50e29i)T^{2} \) |
| 97 | \( 1 + 6.12e14T + 6.33e29T^{2} \) |
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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
−12.38783794304918156585698931840, −11.20712475978009374107813735822, −9.546175044487074263446831075097, −8.431159458872298431420587975054, −7.40471324822829465297055162792, −6.37582329375467881672405746784, −5.54355060400282889774659329704, −3.47238019769086421630314535720, −1.51088094890747016648231391764, −0.39127423967316395629271123987,
0.953693448517366008855852038501, 2.27318062963869853112621107648, 3.55890872073261592706527930451, 5.01692135946025248989469475262, 6.37423182988332132312102098948, 8.194333802445608134361870996922, 9.624830884716461460655310220635, 10.25929643352686225285928424217, 11.15070815883304284811757058048, 12.05858185181063103604915336542
