| L(s) = 1 | + (0.330 − 1.37i)2-s + (−0.989 − 0.142i)3-s + (−1.78 − 0.907i)4-s + (−1.14 − 3.89i)5-s + (−0.522 + 1.31i)6-s + (1.34 + 1.55i)7-s + (−1.83 + 2.15i)8-s + (0.959 + 0.281i)9-s + (−5.72 + 0.286i)10-s + (−2.51 − 1.61i)11-s + (1.63 + 1.15i)12-s + (−2.51 + 2.90i)13-s + (2.58 − 1.33i)14-s + (0.577 + 4.01i)15-s + (2.35 + 3.23i)16-s + (−1.50 + 0.687i)17-s + ⋯ |
| L(s) = 1 | + (0.233 − 0.972i)2-s + (−0.571 − 0.0821i)3-s + (−0.891 − 0.453i)4-s + (−0.510 − 1.73i)5-s + (−0.213 + 0.536i)6-s + (0.509 + 0.587i)7-s + (−0.649 + 0.760i)8-s + (0.319 + 0.0939i)9-s + (−1.81 + 0.0905i)10-s + (−0.757 − 0.486i)11-s + (0.471 + 0.332i)12-s + (−0.697 + 0.804i)13-s + (0.690 − 0.357i)14-s + (0.148 + 1.03i)15-s + (0.587 + 0.809i)16-s + (−0.365 + 0.166i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.132095 + 0.679922i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.132095 + 0.679922i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.330 + 1.37i)T \) |
| 3 | \( 1 + (0.989 + 0.142i)T \) |
| 23 | \( 1 + (2.68 + 3.97i)T \) |
| good | 5 | \( 1 + (1.14 + 3.89i)T + (-4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (-1.34 - 1.55i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (2.51 + 1.61i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (2.51 - 2.90i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.50 - 0.687i)T + (11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.38 + 5.22i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.730 + 1.60i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-2.34 + 0.337i)T + (29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-3.23 + 11.0i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-5.13 + 1.50i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (0.145 - 1.01i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 2.51iT - 47T^{2} \) |
| 53 | \( 1 + (-6.29 + 5.45i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-6.47 - 5.60i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (14.5 - 2.08i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (4.48 - 2.88i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (5.26 + 8.19i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (0.258 - 0.565i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-10.9 + 12.5i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-3.68 - 1.08i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (2.97 + 0.427i)T + (85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-2.39 - 8.15i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−11.68169245349744911306050236445, −10.70588020494521645717421911228, −9.338472863025516544105375401017, −8.791187645564510862489461067305, −7.74944387274167379927113605445, −5.77656619882201018255159726812, −4.91696281829929739979286263525, −4.29237837611051073814480794725, −2.21576089078910807206179396185, −0.52424018767073079446162077133,
3.06999965219826172255973423501, 4.30298691926128678217908715172, 5.52298903863659953473413451764, 6.59427788900045449545355864998, 7.61288531915391812159158706453, 7.81339735001792150700204097053, 9.894304831827151279614235039199, 10.39442732278055156146037140711, 11.47297173240499072712303678649, 12.36934840982300587902527683671
