| L(s) = 1 | + (2.48 − 0.393i)2-s + (0.891 + 0.453i)3-s + (4.11 − 1.33i)4-s + (−2.19 − 0.451i)5-s + (2.39 + 0.777i)6-s + (2.30 + 1.29i)7-s + (5.22 − 2.66i)8-s + (0.587 + 0.809i)9-s + (−5.62 − 0.258i)10-s + (−1.84 − 1.34i)11-s + (4.27 + 0.677i)12-s + (0.560 − 3.53i)13-s + (6.24 + 2.31i)14-s + (−1.74 − 1.39i)15-s + (4.93 − 3.58i)16-s + (−1.87 + 0.956i)17-s + ⋯ |
| L(s) = 1 | + (1.75 − 0.278i)2-s + (0.514 + 0.262i)3-s + (2.05 − 0.669i)4-s + (−0.979 − 0.201i)5-s + (0.976 + 0.317i)6-s + (0.871 + 0.490i)7-s + (1.84 − 0.941i)8-s + (0.195 + 0.269i)9-s + (−1.77 − 0.0818i)10-s + (−0.557 − 0.404i)11-s + (1.23 + 0.195i)12-s + (0.155 − 0.981i)13-s + (1.66 + 0.618i)14-s + (−0.450 − 0.360i)15-s + (1.23 − 0.896i)16-s + (−0.455 + 0.231i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.02532 - 0.523566i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.02532 - 0.523566i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.891 - 0.453i)T \) |
| 5 | \( 1 + (2.19 + 0.451i)T \) |
| 7 | \( 1 + (-2.30 - 1.29i)T \) |
| good | 2 | \( 1 + (-2.48 + 0.393i)T + (1.90 - 0.618i)T^{2} \) |
| 11 | \( 1 + (1.84 + 1.34i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.560 + 3.53i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (1.87 - 0.956i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (0.528 - 1.62i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.931 - 5.88i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (8.45 - 2.74i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.111 + 0.0361i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.857 + 5.41i)T + (-35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (4.79 + 6.60i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (5.42 + 5.42i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.356 - 0.698i)T + (-27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-2.92 + 5.74i)T + (-31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (1.03 - 0.752i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-6.45 + 8.88i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-0.876 - 1.72i)T + (-39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (-2.96 - 9.13i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (9.91 - 1.57i)T + (69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (7.28 - 2.36i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-5.77 - 11.3i)T + (-48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-11.5 - 8.36i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-3.57 + 7.01i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−11.18632530607350965832549674573, −10.44840153075005654764272072637, −8.857976964676366722348445121943, −7.985752867256042386834341572959, −7.15998449062601541893825737603, −5.55841482723122342154247061938, −5.16449342562492022337148216104, −3.93085324196104470960360716399, −3.31892445968405252413835452320, −2.00644206791907300936912924837,
2.11223079905299379999648404345, 3.30174919984753375476257264094, 4.42335423339409710631645531334, 4.74600312638161499242320918931, 6.34228845618479110339514535584, 7.16354373297323313853634062899, 7.76086722130682837029117801366, 8.783766332569973249675342656326, 10.42453743978427683507223491084, 11.50875951732791480181193141936
