| 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 + (−2.82 − 3.26i)7-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (−0.654 + 0.755i)10-s + (4.70 + 3.02i)11-s + (0.841 + 0.540i)12-s + (2.96 − 3.42i)13-s + (−4.14 − 1.21i)14-s + (−0.142 − 0.989i)15-s + (−0.654 − 0.755i)16-s + (−1.15 − 2.52i)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 + (−1.06 − 1.23i)7-s + (−0.0503 − 0.349i)8-s + (−0.319 − 0.0939i)9-s + (−0.207 + 0.238i)10-s + (1.41 + 0.911i)11-s + (0.242 + 0.156i)12-s + (0.822 − 0.948i)13-s + (−1.10 − 0.325i)14-s + (−0.0367 − 0.255i)15-s + (−0.163 − 0.188i)16-s + (−0.279 − 0.612i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.25420 - 1.07222i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25420 - 1.07222i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (3.33 + 3.45i)T \) |
| good | 7 | \( 1 + (2.82 + 3.26i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-4.70 - 3.02i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.96 + 3.42i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.15 + 2.52i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.59 + 5.67i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (2.15 + 4.72i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.711 - 4.94i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (8.32 + 2.44i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-5.69 + 1.67i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.737 + 5.12i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + (-6.49 - 7.49i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (8.14 - 9.39i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.291 - 2.03i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-4.66 + 2.99i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-3.40 + 2.19i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (6.43 - 14.0i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (5.44 - 6.27i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-7.40 - 2.17i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.331 + 2.30i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (11.3 - 3.33i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.44148440608236224049223205873, −9.612701874708975973582146980305, −8.858872668552261124607157064770, −7.25772640686658996306607670296, −6.80549807887229963921534552445, −5.68162322333696866591669110177, −4.28238266568351405910310043928, −3.92432574364548773150851981798, −2.85357863986765118428740889786, −0.76473765892502891669909313457,
1.71558230344387289600322445486, 3.34814533309937042298410786418, 3.93633224787391004498130735482, 5.68593258153637636097262681575, 6.12700216321863806130230778566, 6.82843005007061505863542570147, 8.067455864535078624297313844951, 8.855714300639012317781662288054, 9.460608665684416770864114363396, 11.01971072928409879281338524517
