| L(s) = 1 | + (1.04 + 1.20i)2-s + (−0.198 − 0.127i)3-s + (−0.0771 + 0.536i)4-s + (−0.0536 − 0.373i)6-s + (−0.874 − 0.256i)7-s + (1.95 − 1.25i)8-s + (−1.22 − 2.67i)9-s + (2.87 − 3.32i)11-s + (0.0839 − 0.0968i)12-s + (3.55 − 1.04i)13-s + (−0.603 − 1.32i)14-s + (4.59 + 1.34i)16-s + (−0.0287 − 0.199i)17-s + (1.94 − 4.26i)18-s + (−0.498 + 3.46i)19-s + ⋯ |
| L(s) = 1 | + (0.738 + 0.852i)2-s + (−0.114 − 0.0738i)3-s + (−0.0385 + 0.268i)4-s + (−0.0219 − 0.152i)6-s + (−0.330 − 0.0970i)7-s + (0.691 − 0.444i)8-s + (−0.407 − 0.892i)9-s + (0.868 − 1.00i)11-s + (0.0242 − 0.0279i)12-s + (0.986 − 0.289i)13-s + (−0.161 − 0.353i)14-s + (1.14 + 0.337i)16-s + (−0.00697 − 0.0485i)17-s + (0.459 − 1.00i)18-s + (−0.114 + 0.794i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.15510 + 0.102087i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.15510 + 0.102087i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 + (3.35 - 3.42i)T \) |
| good | 2 | \( 1 + (-1.04 - 1.20i)T + (-0.284 + 1.97i)T^{2} \) |
| 3 | \( 1 + (0.198 + 0.127i)T + (1.24 + 2.72i)T^{2} \) |
| 7 | \( 1 + (0.874 + 0.256i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-2.87 + 3.32i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-3.55 + 1.04i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.0287 + 0.199i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.498 - 3.46i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.339 - 2.36i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-3.00 + 1.93i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (2.46 + 5.39i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (1.56 - 3.42i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-4.40 - 2.83i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 8.39T + 47T^{2} \) |
| 53 | \( 1 + (4.09 + 1.20i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-2.96 + 0.870i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (1.20 - 0.771i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (3.65 + 4.21i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-0.868 - 1.00i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (1.41 - 9.84i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (12.4 - 3.65i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-0.397 - 0.869i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-9.64 - 6.20i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (1.94 - 4.25i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.84070953301991334089231002780, −9.805964362213152311853142284109, −8.854126779795833287037170452243, −7.933035338641574970466677184546, −6.74935416755294901524262698428, −6.08444480097595683956556539816, −5.60536754132923407900963632226, −4.03125335224695605168059191866, −3.43017787596981569589956587326, −1.13968128139571768893057999924,
1.76370950549467882478418550151, 2.83965578099566627702539953858, 4.09127644868851141114155105830, 4.73495040131045135495125237544, 5.97341602066904027733220073544, 7.02646527928007076536387554780, 8.137861313848133117005001547654, 9.036508901632054797356714294455, 10.20075683251061255658504207574, 10.85355328908722657169850922423
