| L(s) = 1 | + (1.24 − 0.664i)2-s + (1.62 + 0.604i)3-s + (1.11 − 1.65i)4-s + (−0.345 + 2.20i)5-s + (2.42 − 0.323i)6-s − 0.309i·7-s + (0.292 − 2.81i)8-s + (2.26 + 1.96i)9-s + (1.03 + 2.98i)10-s + (−1.53 + 1.11i)11-s + (2.81 − 2.01i)12-s + (−3.79 − 2.75i)13-s + (−0.205 − 0.386i)14-s + (−1.89 + 3.37i)15-s + (−1.50 − 3.70i)16-s + (−4.30 + 1.39i)17-s + ⋯ |
| L(s) = 1 | + (0.882 − 0.469i)2-s + (0.937 + 0.349i)3-s + (0.558 − 0.829i)4-s + (−0.154 + 0.987i)5-s + (0.991 − 0.131i)6-s − 0.117i·7-s + (0.103 − 0.994i)8-s + (0.756 + 0.654i)9-s + (0.327 + 0.944i)10-s + (−0.463 + 0.336i)11-s + (0.813 − 0.582i)12-s + (−1.05 − 0.764i)13-s + (−0.0550 − 0.103i)14-s + (−0.489 + 0.871i)15-s + (−0.376 − 0.926i)16-s + (−1.04 + 0.339i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.190i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.58495 - 0.248503i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.58495 - 0.248503i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.24 + 0.664i)T \) |
| 3 | \( 1 + (-1.62 - 0.604i)T \) |
| 5 | \( 1 + (0.345 - 2.20i)T \) |
| good | 7 | \( 1 + 0.309iT - 7T^{2} \) |
| 11 | \( 1 + (1.53 - 1.11i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (3.79 + 2.75i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (4.30 - 1.39i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-5.24 + 1.70i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-5.20 + 3.78i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (6.49 + 2.11i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (5.73 - 1.86i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.24 - 2.35i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (0.902 - 1.24i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 3.75iT - 43T^{2} \) |
| 47 | \( 1 + (-2.19 + 6.75i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.721 - 0.234i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-11.0 - 8.03i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (7.66 - 5.57i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (6.37 - 2.07i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-0.750 + 2.30i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (7.60 - 5.52i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.81 - 0.588i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (0.792 + 2.43i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-5.28 - 7.27i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-3.88 + 11.9i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.67032318739506259911520796126, −10.69049864734287844890741707079, −10.14116518613123421837275061790, −9.139774332926441344258096257468, −7.54046244885941932084789957788, −7.01613809744112337541217629025, −5.41511671950225411621130238426, −4.31238168666368058970937886331, −3.10869225594194918305397909008, −2.34892015567894287964001364968,
2.07399131651978770023719359603, 3.46912367501344172948266649590, 4.63783676678336460817776812436, 5.61052733202164652023855543753, 7.18509661811639867402722249253, 7.63480642317764978053640949227, 8.888355736539557720850292844261, 9.417044084249280260524824842717, 11.23967588707253397746079654477, 12.10534357845249442351835162955
