| L(s) = 1 | + (0.590 − 0.703i)2-s + (0.605 − 0.508i)3-s + (0.200 + 1.13i)4-s − 0.725i·6-s + (−3.19 − 1.16i)7-s + (2.50 + 1.44i)8-s + (−0.412 + 2.33i)9-s + (−1.45 + 2.52i)11-s + (0.700 + 0.587i)12-s + (0.612 − 0.108i)13-s + (−2.70 + 1.56i)14-s + (0.324 − 0.118i)16-s + (5.41 + 0.955i)17-s + (1.40 + 1.67i)18-s + (3.15 + 3.76i)19-s + ⋯ |
| L(s) = 1 | + (0.417 − 0.497i)2-s + (0.349 − 0.293i)3-s + (0.100 + 0.569i)4-s − 0.296i·6-s + (−1.20 − 0.439i)7-s + (0.887 + 0.512i)8-s + (−0.137 + 0.779i)9-s + (−0.438 + 0.759i)11-s + (0.202 + 0.169i)12-s + (0.169 − 0.0299i)13-s + (−0.722 + 0.417i)14-s + (0.0811 − 0.0295i)16-s + (1.31 + 0.231i)17-s + (0.330 + 0.393i)18-s + (0.724 + 0.863i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.79234 + 0.678539i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.79234 + 0.678539i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 37 | \( 1 + (4.16 + 4.43i)T \) |
| good | 2 | \( 1 + (-0.590 + 0.703i)T + (-0.347 - 1.96i)T^{2} \) |
| 3 | \( 1 + (-0.605 + 0.508i)T + (0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (3.19 + 1.16i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (1.45 - 2.52i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.612 + 0.108i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-5.41 - 0.955i)T + (15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (-3.15 - 3.76i)T + (-3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (3.86 - 2.22i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-8.86 - 5.11i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 6.23iT - 31T^{2} \) |
| 41 | \( 1 + (0.0934 + 0.530i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 - 9.30iT - 43T^{2} \) |
| 47 | \( 1 + (-0.483 - 0.837i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.46 - 1.62i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (4.29 + 11.8i)T + (-45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-4.37 + 0.770i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (10.9 + 3.99i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-0.548 + 0.460i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 - 5.46T + 73T^{2} \) |
| 79 | \( 1 + (2.49 - 6.84i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (0.0550 - 0.312i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-3.44 - 9.46i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-10.7 + 6.18i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.18124852052924164040717016686, −9.601910415032704965396824957398, −8.169526532815955354373288505182, −7.75912080369793725207525217157, −6.95834320728891023118191893500, −5.76402849056334795646010025670, −4.67378502841919715159799535317, −3.53654555767653454140788681528, −2.93738827947878929550824727062, −1.70633141109723945834527702524,
0.77593837254425428554256288154, 2.81373937088246968270492899831, 3.52599159403333377821538398914, 4.81772753074249934150908562669, 5.81800845369322278921550967419, 6.32875028037601557932816260304, 7.19363351288339971186546777424, 8.399878580791546023681765514412, 9.193869440566421774176331706781, 10.05088415176110365979404210717
