| L(s) = 1 | + (1.22 + 0.707i)2-s + (−0.598 − 2.93i)3-s + (0.999 + 1.73i)4-s + (5.32 + 3.07i)5-s + (1.34 − 4.02i)6-s + (−4.69 − 5.19i)7-s + 2.82i·8-s + (−8.28 + 3.52i)9-s + (4.34 + 7.52i)10-s + (−15.5 + 8.99i)11-s + (4.49 − 3.97i)12-s + 1.38·13-s + (−2.07 − 9.68i)14-s + (5.84 − 17.4i)15-s + (−2.00 + 3.46i)16-s + (5.32 − 3.07i)17-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.199 − 0.979i)3-s + (0.249 + 0.433i)4-s + (1.06 + 0.614i)5-s + (0.224 − 0.670i)6-s + (−0.670 − 0.742i)7-s + 0.353i·8-s + (−0.920 + 0.391i)9-s + (0.434 + 0.752i)10-s + (−1.41 + 0.818i)11-s + (0.374 − 0.331i)12-s + 0.106·13-s + (−0.147 − 0.691i)14-s + (0.389 − 1.16i)15-s + (−0.125 + 0.216i)16-s + (0.313 − 0.180i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0485i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.41163 - 0.0342894i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41163 - 0.0342894i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.22 - 0.707i)T \) |
| 3 | \( 1 + (0.598 + 2.93i)T \) |
| 7 | \( 1 + (4.69 + 5.19i)T \) |
| good | 5 | \( 1 + (-5.32 - 3.07i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (15.5 - 8.99i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 1.38T + 169T^{2} \) |
| 17 | \( 1 + (-5.32 + 3.07i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-7.53 + 13.0i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-22.9 - 13.2i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 19.0iT - 841T^{2} \) |
| 31 | \( 1 + (9.22 + 15.9i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 11.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 54.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + (46.4 + 26.8i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (35.4 - 20.4i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-47.2 + 27.2i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-35.8 + 62.1i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (10.9 + 19.0i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 109. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-7.11 - 12.3i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (46.3 - 80.3i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 35.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (138. + 80.1i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 118.T + 9.40e3T^{2} \) |
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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
−15.71233147228831871222947757319, −14.29223785676909460186764313983, −13.33882153402943243982040589551, −12.84852111196815134940220686675, −11.11287501863076097816279349973, −9.819112366274895406771443049500, −7.61336623382496843489784072779, −6.70092201068603518595443136175, −5.39911655415851667151436323962, −2.66180964612968656785158117705,
3.01201525965927102144182932257, 5.19748792837762772582726058973, 5.90008394344139750519219699834, 8.763981428634776683221648251980, 9.875161399949021070686978813144, 10.87358697826480611572413113561, 12.45328603420141268510003946390, 13.37171828440895740660002432614, 14.62292155199889268133755608764, 15.91640643866251212942869270753
