| L(s) = 1 | + (0.841 − 0.540i)2-s + (0.142 − 0.989i)3-s + (0.415 − 0.909i)4-s + (−0.959 + 0.281i)5-s + (−0.415 − 0.909i)6-s + (1.54 + 1.78i)7-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (−0.654 + 0.755i)10-s + (−1.62 − 1.04i)11-s + (−0.841 − 0.540i)12-s + (3.97 − 4.58i)13-s + (2.26 + 0.666i)14-s + (0.142 + 0.989i)15-s + (−0.654 − 0.755i)16-s + (−2.45 − 5.36i)17-s + ⋯ |
| L(s) = 1 | + (0.594 − 0.382i)2-s + (0.0821 − 0.571i)3-s + (0.207 − 0.454i)4-s + (−0.429 + 0.125i)5-s + (−0.169 − 0.371i)6-s + (0.585 + 0.675i)7-s + (−0.0503 − 0.349i)8-s + (−0.319 − 0.0939i)9-s + (−0.207 + 0.238i)10-s + (−0.490 − 0.315i)11-s + (−0.242 − 0.156i)12-s + (1.10 − 1.27i)13-s + (0.606 + 0.178i)14-s + (0.0367 + 0.255i)15-s + (−0.163 − 0.188i)16-s + (−0.594 − 1.30i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.133 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.133 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.34883 - 1.54282i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34883 - 1.54282i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 + 0.540i)T \) |
| 3 | \( 1 + (-0.142 + 0.989i)T \) |
| 5 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (-2.20 - 4.25i)T \) |
| good | 7 | \( 1 + (-1.54 - 1.78i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (1.62 + 1.04i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-3.97 + 4.58i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.45 + 5.36i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-3.60 + 7.89i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.26 - 4.95i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (0.123 + 0.861i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (5.29 + 1.55i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (4.21 - 1.23i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (1.59 - 11.0i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 0.306T + 47T^{2} \) |
| 53 | \( 1 + (-4.65 - 5.37i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-2.26 + 2.61i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.80 - 12.5i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-8.25 + 5.30i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-3.04 + 1.95i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.63 + 3.57i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (3.55 - 4.10i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (3.72 + 1.09i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (1.50 - 10.4i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-4.16 + 1.22i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.60660065836946468680032953020, −9.246271969317764108917211478765, −8.507793065376691687817091375545, −7.54311063585947664199889994209, −6.70100975434241992515756455344, −5.43996507811904521200027324433, −4.96197634732334465947122212458, −3.29288318957114085472173456232, −2.64205147729123431851224193502, −0.959184080015194700616323235944,
1.81816075832112091029586533276, 3.68852862687679701218058341820, 4.10488385024698115648251423283, 5.10884298854836982677936772946, 6.20300804737870146009320299796, 7.12821854035448993545838700538, 8.243705240632917971286875548284, 8.607889293698249561653987055572, 10.04441724249633105552293427119, 10.72349311016678287922844644536
