| L(s) = 1 | + (−3.11 − 1.58i)2-s + (2.23 − 2.12i)3-s + (4.83 + 6.62i)4-s + (6.48 − 0.0692i)5-s + (−10.3 + 3.06i)6-s + (6.00 − 3.59i)7-s + (−2.34 − 14.4i)8-s + (0.0275 − 0.515i)9-s + (−20.2 − 10.0i)10-s + (−0.864 + 0.768i)11-s + (24.8 + 4.57i)12-s + (−1.84 − 1.11i)13-s + (−24.3 + 1.69i)14-s + (14.3 − 13.9i)15-s + (−5.63 + 17.5i)16-s + (−1.05 + 0.124i)17-s + ⋯ |
| L(s) = 1 | + (−1.55 − 0.791i)2-s + (0.746 − 0.707i)3-s + (1.20 + 1.65i)4-s + (1.29 − 0.0138i)5-s + (−1.72 + 0.511i)6-s + (0.858 − 0.513i)7-s + (−0.292 − 1.81i)8-s + (0.00306 − 0.0572i)9-s + (−2.02 − 1.00i)10-s + (−0.0786 + 0.0698i)11-s + (2.07 + 0.380i)12-s + (−0.141 − 0.0860i)13-s + (−1.74 + 0.120i)14-s + (0.957 − 0.927i)15-s + (−0.352 + 1.09i)16-s + (−0.0621 + 0.00733i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0291 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0291 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.01371 - 0.984611i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01371 - 0.984611i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-6.00 + 3.59i)T \) |
| good | 2 | \( 1 + (3.11 + 1.58i)T + (2.35 + 3.23i)T^{2} \) |
| 3 | \( 1 + (-2.23 + 2.12i)T + (0.480 - 8.98i)T^{2} \) |
| 5 | \( 1 + (-6.48 + 0.0692i)T + (24.9 - 0.534i)T^{2} \) |
| 11 | \( 1 + (0.864 - 0.768i)T + (14.1 - 120. i)T^{2} \) |
| 13 | \( 1 + (1.84 + 1.11i)T + (78.1 + 149. i)T^{2} \) |
| 17 | \( 1 + (1.05 - 0.124i)T + (281. - 67.3i)T^{2} \) |
| 19 | \( 1 + (-14.7 + 8.48i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-9.23 - 26.8i)T + (-417. + 325. i)T^{2} \) |
| 29 | \( 1 + (-27.3 + 5.33i)T + (779. - 315. i)T^{2} \) |
| 31 | \( 1 + (-27.5 + 10.8i)T + (704. - 653. i)T^{2} \) |
| 37 | \( 1 + (36.0 + 15.5i)T + (941. + 993. i)T^{2} \) |
| 41 | \( 1 + (-6.39 + 4.46i)T + (580. - 1.57e3i)T^{2} \) |
| 43 | \( 1 + (1.33 - 8.24i)T + (-1.75e3 - 582. i)T^{2} \) |
| 47 | \( 1 + (78.7 + 19.7i)T + (1.94e3 + 1.04e3i)T^{2} \) |
| 53 | \( 1 + (11.3 - 41.5i)T + (-2.41e3 - 1.43e3i)T^{2} \) |
| 59 | \( 1 + (28.4 - 13.3i)T + (2.22e3 - 2.67e3i)T^{2} \) |
| 61 | \( 1 + (4.38 + 102. i)T + (-3.70e3 + 317. i)T^{2} \) |
| 67 | \( 1 + (7.70 - 1.16i)T + (4.28e3 - 1.32e3i)T^{2} \) |
| 71 | \( 1 + (-14.3 - 2.79i)T + (4.67e3 + 1.89e3i)T^{2} \) |
| 73 | \( 1 + (-61.5 + 15.4i)T + (4.69e3 - 2.51e3i)T^{2} \) |
| 79 | \( 1 + (0.994 + 13.2i)T + (-6.17e3 + 930. i)T^{2} \) |
| 83 | \( 1 + (94.4 - 9.10i)T + (6.76e3 - 1.31e3i)T^{2} \) |
| 89 | \( 1 + (98.5 - 41.1i)T + (5.57e3 - 5.63e3i)T^{2} \) |
| 97 | \( 1 + (78.3 + 62.5i)T + (2.09e3 + 9.17e3i)T^{2} \) |
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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
−10.83981811009039734086907297739, −9.978177913277696989377737063225, −9.309713050783732800825051460255, −8.352235331591549323254297416159, −7.68509025918164482786872918573, −6.81167001758746384754473346243, −5.14082119479938136880814479189, −3.01123000938212815330902846443, −1.98429586678079707813475782991, −1.22546237794393927575363617950,
1.42116635016265845838147544806, 2.71693522038261341484292671285, 4.85777732776672477257405895306, 5.96113658204174662621096420615, 6.88645358125108655393247201584, 8.324977826147888440717082346253, 8.611282090294505498439765928962, 9.661955481348528436872986546509, 9.992401312991928768852630915268, 10.94116978477076505126111910425
