| L(s) = 1 | + (−0.0713 + 0.997i)2-s + (−3.05 + 0.664i)3-s + (−0.989 − 0.142i)4-s + (0.783 + 2.09i)5-s + (−0.444 − 3.09i)6-s + (−2.28 + 1.24i)7-s + (0.212 − 0.977i)8-s + (6.15 − 2.81i)9-s + (−2.14 + 0.631i)10-s + (−3.47 − 3.01i)11-s + (3.11 − 0.222i)12-s + (1.24 − 2.28i)13-s + (−1.08 − 2.36i)14-s + (−3.78 − 5.87i)15-s + (0.959 + 0.281i)16-s + (−2.99 − 2.24i)17-s + ⋯ |
| L(s) = 1 | + (−0.0504 + 0.705i)2-s + (−1.76 + 0.383i)3-s + (−0.494 − 0.0711i)4-s + (0.350 + 0.936i)5-s + (−0.181 − 1.26i)6-s + (−0.863 + 0.471i)7-s + (0.0751 − 0.345i)8-s + (2.05 − 0.937i)9-s + (−0.678 + 0.199i)10-s + (−1.04 − 0.908i)11-s + (0.899 − 0.0643i)12-s + (0.346 − 0.633i)13-s + (−0.288 − 0.632i)14-s + (−0.976 − 1.51i)15-s + (0.239 + 0.0704i)16-s + (−0.727 − 0.544i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0362390 - 0.0469260i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0362390 - 0.0469260i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0713 - 0.997i)T \) |
| 5 | \( 1 + (-0.783 - 2.09i)T \) |
| 23 | \( 1 + (-0.785 - 4.73i)T \) |
| good | 3 | \( 1 + (3.05 - 0.664i)T + (2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (2.28 - 1.24i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (3.47 + 3.01i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.24 + 2.28i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (2.99 + 2.24i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-0.215 + 1.49i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (6.04 - 0.869i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-4.50 + 2.89i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (8.17 + 3.04i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (4.39 - 9.61i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (0.143 + 0.661i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-1.92 - 1.92i)T + 47iT^{2} \) |
| 53 | \( 1 + (3.01 + 5.52i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-1.80 - 6.13i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (4.00 + 6.23i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (7.74 + 0.554i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (8.49 + 9.79i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (0.891 + 1.19i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (4.18 - 1.22i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (0.858 - 2.30i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (-13.5 - 8.73i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (2.56 + 6.88i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−13.01964690124293121674511300158, −11.67944130079259945635746679757, −10.90527170190746503224146203258, −10.18249112523354261931109207661, −9.237419524816686878731328237829, −7.53694363928120307868267877590, −6.46042263737152231743414800003, −5.87293995782427481870626577873, −5.08542320608782439169660253678, −3.28855990957712180259336834566,
0.05972176614707039496505172027, 1.76537689486996707417662425240, 4.25959552772184238295572623902, 5.14064662792482248788500536982, 6.22097677812018633579540722890, 7.24077255943838136901871012904, 8.794762883595927329094990185444, 10.16764932475736413458763825575, 10.43866055688019490187218923545, 11.67637052404112311624447575670
