| L(s) = 1 | + (0.0713 − 0.997i)2-s + (0.846 − 0.184i)3-s + (−0.989 − 0.142i)4-s + (0.896 + 2.04i)5-s + (−0.123 − 0.857i)6-s + (3.89 − 2.12i)7-s + (−0.212 + 0.977i)8-s + (−2.04 + 0.934i)9-s + (2.10 − 0.748i)10-s + (0.483 + 0.419i)11-s + (−0.863 + 0.0617i)12-s + (0.634 − 1.16i)13-s + (−1.84 − 4.04i)14-s + (1.13 + 1.56i)15-s + (0.959 + 0.281i)16-s + (1.30 + 0.980i)17-s + ⋯ |
| L(s) = 1 | + (0.0504 − 0.705i)2-s + (0.488 − 0.106i)3-s + (−0.494 − 0.0711i)4-s + (0.400 + 0.916i)5-s + (−0.0503 − 0.349i)6-s + (1.47 − 0.804i)7-s + (−0.0751 + 0.345i)8-s + (−0.682 + 0.311i)9-s + (0.666 − 0.236i)10-s + (0.145 + 0.126i)11-s + (−0.249 + 0.0178i)12-s + (0.175 − 0.322i)13-s + (−0.493 − 1.07i)14-s + (0.293 + 0.404i)15-s + (0.239 + 0.0704i)16-s + (0.317 + 0.237i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.46086 - 0.574267i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46086 - 0.574267i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.896 - 2.04i)T \) |
| 23 | \( 1 + (4.44 + 1.79i)T \) |
| good | 3 | \( 1 + (-0.846 + 0.184i)T + (2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (-3.89 + 2.12i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-0.483 - 0.419i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.634 + 1.16i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-1.30 - 0.980i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-0.660 + 4.59i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (2.76 - 0.397i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (0.176 - 0.113i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (8.78 + 3.27i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (0.793 - 1.73i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.66 - 12.2i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-5.65 - 5.65i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.58 - 6.56i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (0.421 + 1.43i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (2.82 + 4.40i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (10.3 + 0.743i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (5.23 + 6.03i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (3.39 + 4.53i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (0.0648 - 0.0190i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (2.38 - 6.38i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (0.562 + 0.361i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (6.52 + 17.5i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−11.82516361674580038579440152218, −10.96195351169034723646568510624, −10.54475551311244938572894498254, −9.250983332930599962656394265871, −8.130277720049080904460208112493, −7.35275877604455861811278038007, −5.75677588228255820169640094735, −4.45413746022156328035015824236, −3.06494553070319981572355037164, −1.80327375364096035205468053333,
1.88028691838139407587713470880, 3.93066377652000463263319169461, 5.30340783059001872000187073477, 5.82928796802637220182388591255, 7.60907518354663852369785020949, 8.629520910300011974377706959310, 8.821933930760241271444115499473, 10.12089903275418640033231888546, 11.75260643287160088534162293178, 12.14672727769764591564477778871
