| L(s) = 1 | + (−0.324 + 0.783i)2-s + (2.32 + 2.32i)4-s + (6.70 − 1.33i)5-s + (0.886 + 0.176i)7-s + (−5.70 + 2.36i)8-s + (−1.13 + 5.68i)10-s + (−3.73 − 5.59i)11-s + (10.5 − 10.5i)13-s + (−0.425 + 0.637i)14-s + 7.89i·16-s + (14.7 + 8.37i)17-s + (−12.9 + 31.3i)19-s + (18.6 + 12.4i)20-s + (5.59 − 1.11i)22-s + (−15.4 + 10.3i)23-s + ⋯ |
| L(s) = 1 | + (−0.162 + 0.391i)2-s + (0.580 + 0.580i)4-s + (1.34 − 0.266i)5-s + (0.126 + 0.0251i)7-s + (−0.712 + 0.295i)8-s + (−0.113 + 0.568i)10-s + (−0.339 − 0.508i)11-s + (0.812 − 0.812i)13-s + (−0.0304 + 0.0455i)14-s + 0.493i·16-s + (0.870 + 0.492i)17-s + (−0.683 + 1.64i)19-s + (0.932 + 0.622i)20-s + (0.254 − 0.0505i)22-s + (−0.672 + 0.449i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.67094 + 0.721236i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67094 + 0.721236i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 + (-14.7 - 8.37i)T \) |
| good | 2 | \( 1 + (0.324 - 0.783i)T + (-2.82 - 2.82i)T^{2} \) |
| 5 | \( 1 + (-6.70 + 1.33i)T + (23.0 - 9.56i)T^{2} \) |
| 7 | \( 1 + (-0.886 - 0.176i)T + (45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (3.73 + 5.59i)T + (-46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (-10.5 + 10.5i)T - 169iT^{2} \) |
| 19 | \( 1 + (12.9 - 31.3i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (15.4 - 10.3i)T + (202. - 488. i)T^{2} \) |
| 29 | \( 1 + (4.13 + 20.7i)T + (-776. + 321. i)T^{2} \) |
| 31 | \( 1 + (-21.1 + 31.6i)T + (-367. - 887. i)T^{2} \) |
| 37 | \( 1 + (33.3 + 22.2i)T + (523. + 1.26e3i)T^{2} \) |
| 41 | \( 1 + (-3.70 - 0.736i)T + (1.55e3 + 643. i)T^{2} \) |
| 43 | \( 1 + (5.21 + 12.5i)T + (-1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-9.20 + 9.20i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (0.763 - 1.84i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (31.7 - 13.1i)T + (2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-6.92 + 34.8i)T + (-3.43e3 - 1.42e3i)T^{2} \) |
| 67 | \( 1 - 31.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (86.1 + 57.5i)T + (1.92e3 + 4.65e3i)T^{2} \) |
| 73 | \( 1 + (-61.7 + 12.2i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (36.2 + 54.3i)T + (-2.38e3 + 5.76e3i)T^{2} \) |
| 83 | \( 1 + (-39.7 - 16.4i)T + (4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (49.5 + 49.5i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-24.9 - 125. i)T + (-8.69e3 + 3.60e3i)T^{2} \) |
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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
−12.92552961799542461884192279505, −12.00797555557048655030335757155, −10.69774704855784760249342308485, −9.861954256444701300771618337844, −8.477065452911987353184043706406, −7.82011786897404740484363702667, −6.05007521648271124781152628652, −5.79539431908543214580637485804, −3.53031354732969204784285265033, −1.90828692586003527344075630516,
1.57600514695590706756976520856, 2.76933802833227562960917016817, 4.98697444946497420001258945953, 6.21038251884511646259958215152, 6.96291541092399907103830902398, 8.821440018693653701016148387361, 9.776953299835841333522927027642, 10.50451442817056092684660934514, 11.39416049780572430653590238944, 12.55321118638266076769246313438
