| 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
−10.72349311016678287922844644536, −10.04441724249633105552293427119, −8.607889293698249561653987055572, −8.243705240632917971286875548284, −7.12821854035448993545838700538, −6.20300804737870146009320299796, −5.10884298854836982677936772946, −4.10488385024698115648251423283, −3.68852862687679701218058341820, −1.81816075832112091029586533276,
0.959184080015194700616323235944, 2.64205147729123431851224193502, 3.29288318957114085472173456232, 4.96197634732334465947122212458, 5.43996507811904521200027324433, 6.70100975434241992515756455344, 7.54311063585947664199889994209, 8.507793065376691687817091375545, 9.246271969317764108917211478765, 10.60660065836946468680032953020
