| L(s) = 1 | + (−1.72 − 1.37i)2-s + (0.800 + 0.0449i)3-s + (0.642 + 2.81i)4-s + (0.220 + 3.93i)5-s + (−1.32 − 1.18i)6-s + (3.01 + 1.45i)7-s + (0.853 − 1.77i)8-s + (−2.34 − 0.263i)9-s + (5.04 − 7.10i)10-s + (0.430 + 0.684i)11-s + (0.388 + 2.28i)12-s + (−1.33 − 3.82i)13-s + (−3.21 − 6.67i)14-s + 3.15i·15-s + (1.28 − 0.620i)16-s + (3.41 + 0.983i)17-s + ⋯ |
| L(s) = 1 | + (−1.22 − 0.974i)2-s + (0.462 + 0.0259i)3-s + (0.321 + 1.40i)4-s + (0.0987 + 1.75i)5-s + (−0.539 − 0.482i)6-s + (1.14 + 0.549i)7-s + (0.301 − 0.626i)8-s + (−0.780 − 0.0879i)9-s + (1.59 − 2.24i)10-s + (0.129 + 0.206i)11-s + (0.112 + 0.659i)12-s + (−0.371 − 1.06i)13-s + (−0.858 − 1.78i)14-s + 0.815i·15-s + (0.322 − 0.155i)16-s + (0.827 + 0.238i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.698925 + 0.0207733i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.698925 + 0.0207733i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 113 | \( 1 + (-4.73 + 9.51i)T \) |
| good | 2 | \( 1 + (1.72 + 1.37i)T + (0.445 + 1.94i)T^{2} \) |
| 3 | \( 1 + (-0.800 - 0.0449i)T + (2.98 + 0.335i)T^{2} \) |
| 5 | \( 1 + (-0.220 - 3.93i)T + (-4.96 + 0.559i)T^{2} \) |
| 7 | \( 1 + (-3.01 - 1.45i)T + (4.36 + 5.47i)T^{2} \) |
| 11 | \( 1 + (-0.430 - 0.684i)T + (-4.77 + 9.91i)T^{2} \) |
| 13 | \( 1 + (1.33 + 3.82i)T + (-10.1 + 8.10i)T^{2} \) |
| 17 | \( 1 + (-3.41 - 0.983i)T + (14.3 + 9.04i)T^{2} \) |
| 19 | \( 1 + (-1.75 + 1.57i)T + (2.12 - 18.8i)T^{2} \) |
| 23 | \( 1 + (3.16 - 3.54i)T + (-2.57 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-2.21 + 7.69i)T + (-24.5 - 15.4i)T^{2} \) |
| 31 | \( 1 + (-3.17 + 1.11i)T + (24.2 - 19.3i)T^{2} \) |
| 37 | \( 1 + (-8.42 - 5.97i)T + (12.2 + 34.9i)T^{2} \) |
| 41 | \( 1 + (-2.38 + 3.79i)T + (-17.7 - 36.9i)T^{2} \) |
| 43 | \( 1 + (3.99 - 2.20i)T + (22.8 - 36.4i)T^{2} \) |
| 47 | \( 1 + (-1.71 + 10.1i)T + (-44.3 - 15.5i)T^{2} \) |
| 53 | \( 1 + (3.59 - 0.820i)T + (47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + (-0.812 - 0.908i)T + (-6.60 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-3.86 + 2.42i)T + (26.4 - 54.9i)T^{2} \) |
| 67 | \( 1 + (6.18 - 1.05i)T + (63.2 - 22.1i)T^{2} \) |
| 71 | \( 1 + (5.36 + 12.9i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (1.51 + 0.626i)T + (51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (3.05 + 4.30i)T + (-26.0 + 74.5i)T^{2} \) |
| 83 | \( 1 + (8.54 - 10.7i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-1.06 + 1.92i)T + (-47.3 - 75.3i)T^{2} \) |
| 97 | \( 1 + (7.27 - 3.50i)T + (60.4 - 75.8i)T^{2} \) |
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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
−13.77894693132213108600306290422, −11.81728776181632487786037650342, −11.46396996850433368149806214975, −10.36468227674411369991455431969, −9.665443278899492257310380123516, −8.191866884118326620868633400000, −7.68605778433644841185441172641, −5.80635014680614386201028954848, −3.17804343083968950065551036596, −2.30016001723301724331972314005,
1.28504260630565055569608168678, 4.54661105182429852377453023424, 5.78221141215919521150476574544, 7.53192480252628803061636146376, 8.315827811459663954899182628824, 8.911563619434302210325938914668, 9.849456275690205251942384466788, 11.42324514331361942844311963544, 12.55433523613852429244527098702, 14.11189622603262893030420294370
