| 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 + (−1.50 − 1.73i)7-s + (0.142 + 0.989i)8-s + (−0.959 − 0.281i)9-s + (−0.654 + 0.755i)10-s + (−0.853 − 0.548i)11-s + (0.841 + 0.540i)12-s + (0.709 − 0.819i)13-s + (2.20 + 0.647i)14-s + (0.142 + 0.989i)15-s + (−0.654 − 0.755i)16-s + (−1.38 − 3.03i)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 + (−0.568 − 0.656i)7-s + (0.0503 + 0.349i)8-s + (−0.319 − 0.0939i)9-s + (−0.207 + 0.238i)10-s + (−0.257 − 0.165i)11-s + (0.242 + 0.156i)12-s + (0.196 − 0.227i)13-s + (0.589 + 0.173i)14-s + (0.0367 + 0.255i)15-s + (−0.163 − 0.188i)16-s + (−0.336 − 0.736i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.707052 - 0.374509i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.707052 - 0.374509i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (4.48 + 1.70i)T \) |
| good | 7 | \( 1 + (1.50 + 1.73i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (0.853 + 0.548i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-0.709 + 0.819i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.38 + 3.03i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.74 + 3.82i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.28 - 5.01i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (1.47 + 10.2i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (9.82 + 2.88i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-7.18 + 2.11i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.0393 + 0.273i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 2.17T + 47T^{2} \) |
| 53 | \( 1 + (-8.32 - 9.60i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-2.87 + 3.31i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (0.477 + 3.31i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (1.08 - 0.696i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-12.1 + 7.81i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-3.01 + 6.59i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (2.02 - 2.33i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (14.2 + 4.18i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (0.698 - 4.85i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-14.1 + 4.15i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.27490692433875040337949907242, −9.417743291462960725121829541617, −8.859323938506297412555917473882, −7.70083652527899406739111056336, −6.86402913285437888770642213107, −5.91132843304411400419318508839, −5.00622714897998700111866275162, −3.81500954847051676008258578533, −2.46526759758476358716721687505, −0.51565585896097122860007920216,
1.56676778007319739461818083314, 2.59836945827446143272772391193, 3.79102497580530418694206120645, 5.44145624082257919871139151804, 6.27070776467175148588385051172, 7.10533670242471346342236297809, 8.205542340372327607326925642637, 8.826839341659125677462927235880, 9.866134260502983348791351034327, 10.40169580311401110648405717277
