| L(s) = 1 | + (64 − 110. i)2-s + (−3.40e3 − 5.89e3i)3-s + (−8.19e3 − 1.41e4i)4-s + (−5.53e4 + 9.58e4i)5-s − 8.70e5·6-s + (5.50e5 + 2.10e6i)7-s − 2.09e6·8-s + (−1.59e7 + 2.76e7i)9-s + (7.08e6 + 1.22e7i)10-s + (−2.69e7 − 4.66e7i)11-s + (−5.57e7 + 9.65e7i)12-s + 2.39e8·13-s + (2.68e8 + 7.38e7i)14-s + 7.53e8·15-s + (−1.34e8 + 2.32e8i)16-s + (−1.29e9 − 2.23e9i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.897 − 1.55i)3-s + (−0.249 − 0.433i)4-s + (−0.316 + 0.548i)5-s − 1.26·6-s + (0.252 + 0.967i)7-s − 0.353·8-s + (−1.11 + 1.92i)9-s + (0.224 + 0.388i)10-s + (−0.417 − 0.722i)11-s + (−0.448 + 0.777i)12-s + 1.05·13-s + (0.681 + 0.187i)14-s + 1.13·15-s + (−0.125 + 0.216i)16-s + (−0.763 − 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(0.694660 + 0.111735i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.694660 + 0.111735i\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-64 + 110. i)T \) |
| 7 | \( 1 + (-5.50e5 - 2.10e6i)T \) |
| good | 3 | \( 1 + (3.40e3 + 5.89e3i)T + (-7.17e6 + 1.24e7i)T^{2} \) |
| 5 | \( 1 + (5.53e4 - 9.58e4i)T + (-1.52e10 - 2.64e10i)T^{2} \) |
| 11 | \( 1 + (2.69e7 + 4.66e7i)T + (-2.08e15 + 3.61e15i)T^{2} \) |
| 13 | \( 1 - 2.39e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + (1.29e9 + 2.23e9i)T + (-1.43e18 + 2.47e18i)T^{2} \) |
| 19 | \( 1 + (2.01e9 - 3.48e9i)T + (-7.59e18 - 1.31e19i)T^{2} \) |
| 23 | \( 1 + (3.30e9 - 5.73e9i)T + (-1.33e20 - 2.30e20i)T^{2} \) |
| 29 | \( 1 - 5.84e9T + 8.62e21T^{2} \) |
| 31 | \( 1 + (-1.50e11 - 2.60e11i)T + (-1.17e22 + 2.03e22i)T^{2} \) |
| 37 | \( 1 + (-7.37e10 + 1.27e11i)T + (-1.66e23 - 2.88e23i)T^{2} \) |
| 41 | \( 1 - 7.07e11T + 1.55e24T^{2} \) |
| 43 | \( 1 + 1.37e12T + 3.17e24T^{2} \) |
| 47 | \( 1 + (1.97e12 - 3.42e12i)T + (-6.03e24 - 1.04e25i)T^{2} \) |
| 53 | \( 1 + (3.72e12 + 6.44e12i)T + (-3.65e25 + 6.33e25i)T^{2} \) |
| 59 | \( 1 + (-1.81e13 - 3.14e13i)T + (-1.82e26 + 3.16e26i)T^{2} \) |
| 61 | \( 1 + (-1.22e13 + 2.12e13i)T + (-3.01e26 - 5.21e26i)T^{2} \) |
| 67 | \( 1 + (6.14e12 + 1.06e13i)T + (-1.23e27 + 2.13e27i)T^{2} \) |
| 71 | \( 1 + 3.63e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + (-4.65e13 - 8.05e13i)T + (-4.45e27 + 7.71e27i)T^{2} \) |
| 79 | \( 1 + (1.41e14 - 2.44e14i)T + (-1.45e28 - 2.52e28i)T^{2} \) |
| 83 | \( 1 + 2.65e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + (2.36e14 - 4.09e14i)T + (-8.70e28 - 1.50e29i)T^{2} \) |
| 97 | \( 1 + 4.86e14T + 6.33e29T^{2} \) |
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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
−15.95669349531761949985966680782, −14.05124134851194002899211121980, −12.91101941648031843348128787599, −11.71460928754713752449217680877, −11.01153937811895123027769630312, −8.354204766315234839734741933364, −6.61887200827526406675853429016, −5.43320923546760774497977949491, −2.75495010258568039289721186533, −1.28965870226837392714517583678,
0.27858372768645209255171147914, 4.05305327805003707879871402865, 4.65328711055828156524064190207, 6.30430387850827832072372123422, 8.486466228508382850045924530782, 10.19577478846712644313994641473, 11.32836441268859964700114325654, 13.08989581241698254199189882084, 14.97399816436708581661411644992, 15.82661025729247561965576179999
