| L(s) = 1 | + (−0.996 + 0.724i)2-s + (0.216 − 2.06i)3-s + (−0.148 + 0.458i)4-s + (0.772 − 1.33i)5-s + (1.27 + 2.21i)6-s + (3.72 − 0.790i)7-s + (−0.944 − 2.90i)8-s + (−1.27 − 0.271i)9-s + (0.199 + 1.89i)10-s + (2.51 + 2.79i)11-s + (0.913 + 0.406i)12-s + (2.40 − 1.07i)13-s + (−3.13 + 3.48i)14-s + (−2.59 − 1.88i)15-s + (2.26 + 1.64i)16-s + (−2.52 + 2.80i)17-s + ⋯ |
| L(s) = 1 | + (−0.704 + 0.512i)2-s + (0.125 − 1.19i)3-s + (−0.0744 + 0.229i)4-s + (0.345 − 0.598i)5-s + (0.521 + 0.903i)6-s + (1.40 − 0.298i)7-s + (−0.334 − 1.02i)8-s + (−0.425 − 0.0904i)9-s + (0.0629 + 0.598i)10-s + (0.758 + 0.842i)11-s + (0.263 + 0.117i)12-s + (0.667 − 0.297i)13-s + (−0.838 + 0.930i)14-s + (−0.669 − 0.486i)15-s + (0.567 + 0.412i)16-s + (−0.612 + 0.680i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.45854 - 0.312260i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45854 - 0.312260i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 + (0.996 - 0.724i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.216 + 2.06i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 + (-0.772 + 1.33i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-3.72 + 0.790i)T + (6.39 - 2.84i)T^{2} \) |
| 11 | \( 1 + (-2.51 - 2.79i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-2.40 + 1.07i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (2.52 - 2.80i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-5.57 - 2.48i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (0.281 + 0.865i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (5.50 - 4.00i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (0.907 + 1.57i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.0352 + 0.335i)T + (-40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (3.54 + 1.57i)T + (28.7 + 31.9i)T^{2} \) |
| 47 | \( 1 + (-0.962 - 0.698i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (2.29 + 0.487i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (0.813 - 7.73i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 - 2.72T + 61T^{2} \) |
| 67 | \( 1 + (-3.71 + 6.42i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.98 - 1.05i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (3.60 + 4.00i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (-6.51 + 7.23i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (0.877 + 8.34i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + (1.57 - 4.84i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-3.37 + 10.3i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−9.631646015053375469542278424958, −8.827547386442388932998241045643, −8.206070196262460242282266091819, −7.49624894255615066522820546221, −6.99361537290854156806461364193, −5.93286793691805000600495225060, −4.76719664861256056202519056826, −3.68987770223656490770238357389, −1.76137000115694517783896238306, −1.19760498993459381065031113114,
1.25118120754862596623467659603, 2.50370061639270252552493513804, 3.72066889298551237816894367995, 4.84734834096519023921686264839, 5.49567126174240933453052035139, 6.62653289412077995976165427421, 7.973677293390664691760373599340, 8.822004559437516743974066929981, 9.327615990073298388907778703112, 10.00205057223140824866486687910
