| 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
−13.71876814780977522997779707734, −12.84741835938370331742207541979, −12.68354803387925853323354887357, −10.42803736028425577962314424602, −9.538283901102038775039050019002, −7.43419275338477083301425013339, −6.82042112798657572131161640938, −6.34070549852126250542813739847, −4.97512938450815742914743567428, −2.42305771792240454310390428813,
2.57861979433938734627110373134, 3.84017169923046951904990859309, 5.27348440186275414197093183051, 6.05504350174916472132525276894, 8.931609230677887674465071760254, 9.883607629904839472395047265257, 10.20389423117839672942204904011, 11.45160528469574853506250611332, 12.65886293586044527913252189480, 13.33091951618459562333815970261
