| L(s) = 1 | + (−1.5 + 0.866i)3-s + (−2.5 + 0.866i)7-s + (1.5 − 2.59i)9-s + (2.5 − 2.59i)13-s + (0.5 − 0.866i)19-s + (3 − 3.46i)21-s + (−2.5 − 4.33i)25-s + 5.19i·27-s + (5.5 + 9.52i)31-s + (4.5 + 2.59i)37-s + (−1.5 + 6.06i)39-s + 13·43-s + (5.5 − 4.33i)49-s + 1.73i·57-s + (6 + 3.46i)61-s + ⋯ |
| L(s) = 1 | + (−0.866 + 0.499i)3-s + (−0.944 + 0.327i)7-s + (0.5 − 0.866i)9-s + (0.693 − 0.720i)13-s + (0.114 − 0.198i)19-s + (0.654 − 0.755i)21-s + (−0.5 − 0.866i)25-s + 0.999i·27-s + (0.987 + 1.71i)31-s + (0.739 + 0.427i)37-s + (−0.240 + 0.970i)39-s + 1.98·43-s + (0.785 − 0.618i)49-s + 0.229i·57-s + (0.768 + 0.443i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1092 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1092 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.028614484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.028614484\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 - 0.866i)T \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
| 13 | \( 1 + (-2.5 + 2.59i)T \) |
| good | 5 | \( 1 + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (-5.5 - 9.52i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.5 - 2.59i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 13T + 43T^{2} \) |
| 47 | \( 1 + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6 - 3.46i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (13.5 - 7.79i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-8.5 - 14.7i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.5 + 14.7i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−10.12169977362830079980874039980, −9.230791737408314168505925757874, −8.451885012156436736158704585555, −7.24910451961876776715859161049, −6.29173855629060028346762143683, −5.83482797122213156267740243274, −4.78515882438589972836099939445, −3.77436050772187512467141989215, −2.79296309236800316745591128435, −0.841386671905321524447549710656,
0.78087809859652073020002816056, 2.21928571692109024015281206773, 3.68012499322436564999616739568, 4.57211991401083487263776655709, 5.91328737160495194495090261104, 6.23052473208453934933233079963, 7.24485864998816846124929316564, 7.88727706244003433389072598492, 9.173230583741711556440882037215, 9.799263031744037377737489341821
