| L(s) = 1 | + (−1.30 + 0.541i)2-s + (0.718 − 3.61i)3-s + (1.41 − 1.41i)4-s + (1.85 − 1.24i)5-s + (1.01 + 5.10i)6-s + (7.14 + 4.77i)7-s + (−1.08 + 2.61i)8-s + (−4.21 − 1.74i)9-s + (−1.75 + 2.62i)10-s + (12.3 − 2.45i)11-s + (−4.09 − 6.12i)12-s + (−8.64 − 8.64i)13-s + (−11.9 − 2.36i)14-s + (−3.15 − 7.60i)15-s − 4i·16-s + (−0.924 − 16.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.653 + 0.270i)2-s + (0.239 − 1.20i)3-s + (0.353 − 0.353i)4-s + (0.371 − 0.248i)5-s + (0.169 + 0.851i)6-s + (1.02 + 0.681i)7-s + (−0.135 + 0.326i)8-s + (−0.467 − 0.193i)9-s + (−0.175 + 0.262i)10-s + (1.12 − 0.223i)11-s + (−0.340 − 0.510i)12-s + (−0.665 − 0.665i)13-s + (−0.850 − 0.169i)14-s + (−0.210 − 0.507i)15-s − 0.250i·16-s + (−0.0543 − 0.998i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 + 0.875i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.482 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.20328 - 0.710719i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20328 - 0.710719i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.30 - 0.541i)T \) |
| 5 | \( 1 + (-1.85 + 1.24i)T \) |
| 17 | \( 1 + (0.924 + 16.9i)T \) |
| good | 3 | \( 1 + (-0.718 + 3.61i)T + (-8.31 - 3.44i)T^{2} \) |
| 7 | \( 1 + (-7.14 - 4.77i)T + (18.7 + 45.2i)T^{2} \) |
| 11 | \( 1 + (-12.3 + 2.45i)T + (111. - 46.3i)T^{2} \) |
| 13 | \( 1 + (8.64 + 8.64i)T + 169iT^{2} \) |
| 19 | \( 1 + (12.5 - 5.21i)T + (255. - 255. i)T^{2} \) |
| 23 | \( 1 + (2.35 + 11.8i)T + (-488. + 202. i)T^{2} \) |
| 29 | \( 1 + (-15.9 - 23.8i)T + (-321. + 776. i)T^{2} \) |
| 31 | \( 1 + (11.5 + 2.29i)T + (887. + 367. i)T^{2} \) |
| 37 | \( 1 + (6.46 - 32.5i)T + (-1.26e3 - 523. i)T^{2} \) |
| 41 | \( 1 + (-48.5 - 32.4i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (39.0 + 16.1i)T + (1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (35.2 + 35.2i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-48.2 + 20.0i)T + (1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (22.2 - 53.6i)T + (-2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (57.2 - 85.7i)T + (-1.42e3 - 3.43e3i)T^{2} \) |
| 67 | \( 1 - 72.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (10.7 - 54.1i)T + (-4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-66.7 + 44.6i)T + (2.03e3 - 4.92e3i)T^{2} \) |
| 79 | \( 1 + (-8.83 + 1.75i)T + (5.76e3 - 2.38e3i)T^{2} \) |
| 83 | \( 1 + (-24.8 - 59.9i)T + (-4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-117. + 117. i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (-19.4 - 29.1i)T + (-3.60e3 + 8.69e3i)T^{2} \) |
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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
−12.22324128433860669685865185340, −11.62736721142323759956413389871, −10.23671438771756365330260329530, −8.952586299023100897632055701210, −8.278775454180827584423511589985, −7.25835839921885268170888280088, −6.26994984119420978854227172253, −4.96800817715015370838929951178, −2.41377670136518099272165039095, −1.21060852128221888290570136122,
1.81570265603721354229795366133, 3.80370607182746552139350737718, 4.65573484242998430733849537897, 6.49739612349005972277564451913, 7.75205950599952001139444061534, 8.996371617596373906905838648968, 9.673504916865596181057737851147, 10.62807492924486185133290262514, 11.25958455814996327833072886049, 12.45856980782153776000567156259
