| L(s) = 1 | + (−0.512 + 0.678i)2-s + (0.835 + 1.67i)3-s + (0.349 + 1.22i)4-s + (−0.121 − 1.31i)5-s + (−1.56 − 0.292i)6-s + (−0.469 + 0.181i)7-s + (−2.59 − 1.00i)8-s + (−0.310 + 0.410i)9-s + (0.952 + 0.589i)10-s + (−1.17 + 1.55i)11-s + (−1.77 + 1.61i)12-s + (3.25 − 1.26i)13-s + (0.117 − 0.411i)14-s + (2.10 − 1.30i)15-s + (−0.160 + 0.0994i)16-s + (3.46 − 0.648i)17-s + ⋯ |
| L(s) = 1 | + (−0.362 + 0.479i)2-s + (0.482 + 0.968i)3-s + (0.174 + 0.614i)4-s + (−0.0544 − 0.587i)5-s + (−0.639 − 0.119i)6-s + (−0.177 + 0.0687i)7-s + (−0.918 − 0.355i)8-s + (−0.103 + 0.136i)9-s + (0.301 + 0.186i)10-s + (−0.353 + 0.467i)11-s + (−0.511 + 0.465i)12-s + (0.902 − 0.349i)13-s + (0.0312 − 0.109i)14-s + (0.542 − 0.336i)15-s + (−0.0401 + 0.0248i)16-s + (0.841 − 0.157i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0286 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0286 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.688133 + 0.708120i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.688133 + 0.708120i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 + (-10.0 + 1.24i)T \) |
| good | 2 | \( 1 + (0.512 - 0.678i)T + (-0.547 - 1.92i)T^{2} \) |
| 3 | \( 1 + (-0.835 - 1.67i)T + (-1.80 + 2.39i)T^{2} \) |
| 5 | \( 1 + (0.121 + 1.31i)T + (-4.91 + 0.918i)T^{2} \) |
| 7 | \( 1 + (0.469 - 0.181i)T + (5.17 - 4.71i)T^{2} \) |
| 11 | \( 1 + (1.17 - 1.55i)T + (-3.01 - 10.5i)T^{2} \) |
| 13 | \( 1 + (-3.25 + 1.26i)T + (9.60 - 8.75i)T^{2} \) |
| 17 | \( 1 + (-3.46 + 0.648i)T + (15.8 - 6.14i)T^{2} \) |
| 19 | \( 1 + (0.388 - 0.780i)T + (-11.4 - 15.1i)T^{2} \) |
| 23 | \( 1 + (4.38 + 5.80i)T + (-6.29 + 22.1i)T^{2} \) |
| 29 | \( 1 + (0.446 + 4.81i)T + (-28.5 + 5.32i)T^{2} \) |
| 31 | \( 1 + (5.11 - 3.16i)T + (13.8 - 27.7i)T^{2} \) |
| 37 | \( 1 + (-0.511 + 0.465i)T + (3.41 - 36.8i)T^{2} \) |
| 41 | \( 1 + (0.120 - 1.30i)T + (-40.3 - 7.53i)T^{2} \) |
| 43 | \( 1 + (-3.31 - 3.02i)T + (3.96 + 42.8i)T^{2} \) |
| 47 | \( 1 + 2.32T + 47T^{2} \) |
| 53 | \( 1 + (1.03 - 2.07i)T + (-31.9 - 42.2i)T^{2} \) |
| 59 | \( 1 + (9.61 + 3.72i)T + (43.6 + 39.7i)T^{2} \) |
| 61 | \( 1 + (11.1 - 2.09i)T + (56.8 - 22.0i)T^{2} \) |
| 67 | \( 1 + (-0.174 - 0.0677i)T + (49.5 + 45.1i)T^{2} \) |
| 71 | \( 1 + (1.31 - 14.2i)T + (-69.7 - 13.0i)T^{2} \) |
| 73 | \( 1 + (-1.01 + 10.9i)T + (-71.7 - 13.4i)T^{2} \) |
| 79 | \( 1 + (1.49 + 16.1i)T + (-77.6 + 14.5i)T^{2} \) |
| 83 | \( 1 + (-4.18 + 1.62i)T + (61.3 - 55.9i)T^{2} \) |
| 89 | \( 1 + (0.350 - 1.23i)T + (-75.6 - 46.8i)T^{2} \) |
| 97 | \( 1 + (16.0 + 2.99i)T + (90.4 + 35.0i)T^{2} \) |
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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
−14.38166889139262426713025220756, −12.88433171653450516287437572005, −12.19310541429801713797292217234, −10.62335697629471829652738211164, −9.524865924582513185390228269932, −8.650294225955697711197649051335, −7.75265315021083207121990362180, −6.18645663052423238590692517104, −4.46252156615337893862432854878, −3.19747912690013940044201845790,
1.63713634627368078038431469860, 3.19696976166733861990133277435, 5.71624996341704338234124888461, 6.86123798884064144469141880419, 8.029627767595038587088827275836, 9.258072479160253646126224685489, 10.49775680241856358619449191133, 11.27656051168414417172699945396, 12.49027398324468929550995661288, 13.63998810799013503068003735674
