| L(s) = 1 | + (−2.52 − 4.37i)3-s − 9.82·5-s + (−17.3 + 30.0i)7-s + (108. − 188. i)9-s + (95.6 + 165. i)11-s + (−607. + 43.0i)13-s + (24.7 + 42.9i)15-s + (−32.0 + 55.4i)17-s + (1.11e3 − 1.92e3i)19-s + 175.·21-s + (1.58e3 + 2.74e3i)23-s − 3.02e3·25-s − 2.32e3·27-s + (332. + 575. i)29-s − 9.73e3·31-s + ⋯ |
| L(s) = 1 | + (−0.161 − 0.280i)3-s − 0.175·5-s + (−0.133 + 0.231i)7-s + (0.447 − 0.775i)9-s + (0.238 + 0.412i)11-s + (−0.997 + 0.0707i)13-s + (0.0284 + 0.0492i)15-s + (−0.0268 + 0.0465i)17-s + (0.707 − 1.22i)19-s + 0.0866·21-s + (0.625 + 1.08i)23-s − 0.969·25-s − 0.613·27-s + (0.0734 + 0.127i)29-s − 1.81·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3217068067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3217068067\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + (607. - 43.0i)T \) |
| good | 3 | \( 1 + (2.52 + 4.37i)T + (-121.5 + 210. i)T^{2} \) |
| 5 | \( 1 + 9.82T + 3.12e3T^{2} \) |
| 7 | \( 1 + (17.3 - 30.0i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-95.6 - 165. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 17 | \( 1 + (32.0 - 55.4i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-1.11e3 + 1.92e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-1.58e3 - 2.74e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-332. - 575. i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + 9.73e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-1.60e3 - 2.77e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + (-2.53e3 - 4.38e3i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (3.95e3 - 6.85e3i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + 1.75e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.37e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (1.56e4 - 2.70e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.48e4 - 2.57e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.40e4 + 2.42e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (2.82e4 - 4.89e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + 4.21e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.88e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.26e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-5.40e4 - 9.36e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (5.22e4 - 9.04e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−11.94530952288592834555991395411, −11.12179518071531203539053430514, −9.629567936726402438217535790648, −9.243369712134051948171634849363, −7.57159676775014331538510493228, −6.96875479171521520327496041134, −5.67223073607828796480046557698, −4.43247703583396093043944438080, −3.04891768068493915013530683212, −1.45549432459558163830457960080,
0.097882809477725642211076763336, 1.88512455544293546569650833338, 3.50326104920358843617273694039, 4.69886170304558602030941799395, 5.76905318905634926748578297335, 7.18973081479906150042820037044, 7.965350269349587591154545364423, 9.313699513381954755469306060266, 10.21080695293045465991614685478, 11.02423667607197201549675672039
