| L(s) = 1 | + (1.00 − 1.73i)2-s + (−0.415 + 2.35i)3-s + (−1.01 − 1.75i)4-s + (3.15 − 1.14i)5-s + (3.68 + 3.09i)6-s + (−3.89 + 1.41i)7-s − 0.0647·8-s + (−2.56 − 0.934i)9-s + (1.17 − 6.63i)10-s + (−0.887 − 5.03i)11-s + (4.57 − 1.66i)12-s + (−3.80 + 1.38i)13-s + (−1.44 + 8.20i)14-s + (1.39 + 7.91i)15-s + (1.96 − 3.40i)16-s + (−1.06 + 1.84i)17-s + ⋯ |
| L(s) = 1 | + (0.709 − 1.22i)2-s + (−0.240 + 1.36i)3-s + (−0.508 − 0.879i)4-s + (1.41 − 0.513i)5-s + (1.50 + 1.26i)6-s + (−1.47 + 0.536i)7-s − 0.0228·8-s + (−0.856 − 0.311i)9-s + (0.370 − 2.09i)10-s + (−0.267 − 1.51i)11-s + (1.31 − 0.480i)12-s + (−1.05 + 0.383i)13-s + (−0.386 + 2.19i)14-s + (0.360 + 2.04i)15-s + (0.491 − 0.851i)16-s + (−0.258 + 0.447i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.35773 - 0.417341i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35773 - 0.417341i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 109 | \( 1 + (-0.968 - 10.3i)T \) |
| good | 2 | \( 1 + (-1.00 + 1.73i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.415 - 2.35i)T + (-2.81 - 1.02i)T^{2} \) |
| 5 | \( 1 + (-3.15 + 1.14i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (3.89 - 1.41i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (0.887 + 5.03i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (3.80 - 1.38i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (1.06 - 1.84i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.09 - 3.63i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.45 - 2.51i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.00 + 5.70i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.15 - 0.419i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-2.14 - 0.779i)T + (28.3 + 23.7i)T^{2} \) |
| 41 | \( 1 + 1.13T + 41T^{2} \) |
| 43 | \( 1 + (-1.23 + 2.13i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.47 + 2.91i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (2.41 - 0.879i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (0.0923 + 0.523i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-7.52 - 6.31i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-5.07 - 1.84i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-6.79 - 11.7i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.129 + 0.733i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (4.33 + 1.57i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (0.0770 - 0.436i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (0.734 + 0.615i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-6.80 + 2.47i)T + (74.3 - 62.3i)T^{2} \) |
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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
−13.33091951618459562333815970261, −12.65886293586044527913252189480, −11.45160528469574853506250611332, −10.20389423117839672942204904011, −9.883607629904839472395047265257, −8.931609230677887674465071760254, −6.05504350174916472132525276894, −5.27348440186275414197093183051, −3.84017169923046951904990859309, −2.57861979433938734627110373134,
2.42305771792240454310390428813, 4.97512938450815742914743567428, 6.34070549852126250542813739847, 6.82042112798657572131161640938, 7.43419275338477083301425013339, 9.538283901102038775039050019002, 10.42803736028425577962314424602, 12.68354803387925853323354887357, 12.84741835938370331742207541979, 13.71876814780977522997779707734
