| L(s) = 1 | + 2.82i·2-s + (8.57 − 2.72i)3-s − 8.00·4-s − 0.912i·5-s + (7.71 + 24.2i)6-s − 19.2·7-s − 22.6i·8-s + (66.1 − 46.7i)9-s + 2.58·10-s − 60.8i·11-s + (−68.6 + 21.8i)12-s + 277.·13-s − 54.5i·14-s + (−2.48 − 7.82i)15-s + 64.0·16-s + 343. i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (0.952 − 0.303i)3-s − 0.500·4-s − 0.0365i·5-s + (0.214 + 0.673i)6-s − 0.393·7-s − 0.353i·8-s + (0.816 − 0.577i)9-s + 0.0258·10-s − 0.503i·11-s + (−0.476 + 0.151i)12-s + 1.64·13-s − 0.278i·14-s + (−0.0110 − 0.0347i)15-s + 0.250·16-s + 1.18i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.585728847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.585728847\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.82iT \) |
| 3 | \( 1 + (-8.57 + 2.72i)T \) |
| 29 | \( 1 + 156. iT \) |
| good | 5 | \( 1 + 0.912iT - 625T^{2} \) |
| 7 | \( 1 + 19.2T + 2.40e3T^{2} \) |
| 11 | \( 1 + 60.8iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 277.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 343. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 240.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 790. iT - 2.79e5T^{2} \) |
| 31 | \( 1 - 695.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 446.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 206. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 850.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 1.68e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.57e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 3.67e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 5.27e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 2.04e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 7.09e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 2.37e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 8.14e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.19e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 7.12e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.31e4T + 8.85e7T^{2} \) |
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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
−12.55606769442444497602036703732, −10.96192842581100399737117588917, −9.819643672180677026913976830835, −8.577813569731570995769251088189, −8.279947143395356587864785697354, −6.79500034440510790297168047810, −6.02883212511090372201944717628, −4.21713397289348765186647019813, −3.08790043706665695858207919658, −1.11063379987957901466150283762,
1.31481579518331799074535440769, 2.89590122240287278432854474710, 3.80474965122014776948633242281, 5.17962671284454113310163108023, 6.92472674871924424251154435054, 8.162043561478312421532540834179, 9.183717691883473957499935257523, 9.848016105411449897672829836536, 10.93941855565900132471582746644, 11.93273623081589957055439257412
