| L(s) = 1 | + (2.59 + 3.25i)2-s + (2.96 − 1.42i)3-s + (−2.08 + 9.11i)4-s + (9.52 − 4.58i)5-s + (12.3 + 5.95i)6-s + (−16.3 + 8.78i)7-s + (−5.06 + 2.43i)8-s + (−10.0 + 12.6i)9-s + (39.6 + 19.1i)10-s + (−34.1 − 42.8i)11-s + (6.85 + 30.0i)12-s + (−18.7 − 23.4i)13-s + (−70.9 − 30.3i)14-s + (21.7 − 27.2i)15-s + (46.3 + 22.2i)16-s + (9.96 + 43.6i)17-s + ⋯ |
| L(s) = 1 | + (0.918 + 1.15i)2-s + (0.571 − 0.275i)3-s + (−0.260 + 1.13i)4-s + (0.851 − 0.410i)5-s + (0.841 + 0.405i)6-s + (−0.880 + 0.474i)7-s + (−0.223 + 0.107i)8-s + (−0.372 + 0.467i)9-s + (1.25 + 0.604i)10-s + (−0.936 − 1.17i)11-s + (0.164 + 0.722i)12-s + (−0.399 − 0.501i)13-s + (−1.35 − 0.578i)14-s + (0.373 − 0.468i)15-s + (0.723 + 0.348i)16-s + (0.142 + 0.622i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.04853 + 1.27876i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.04853 + 1.27876i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (16.3 - 8.78i)T \) |
| good | 2 | \( 1 + (-2.59 - 3.25i)T + (-1.78 + 7.79i)T^{2} \) |
| 3 | \( 1 + (-2.96 + 1.42i)T + (16.8 - 21.1i)T^{2} \) |
| 5 | \( 1 + (-9.52 + 4.58i)T + (77.9 - 97.7i)T^{2} \) |
| 11 | \( 1 + (34.1 + 42.8i)T + (-296. + 1.29e3i)T^{2} \) |
| 13 | \( 1 + (18.7 + 23.4i)T + (-488. + 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-9.96 - 43.6i)T + (-4.42e3 + 2.13e3i)T^{2} \) |
| 19 | \( 1 - 71.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-37.1 + 162. i)T + (-1.09e4 - 5.27e3i)T^{2} \) |
| 29 | \( 1 + (4.18 + 18.3i)T + (-2.19e4 + 1.05e4i)T^{2} \) |
| 31 | \( 1 + 114.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-18.8 - 82.4i)T + (-4.56e4 + 2.19e4i)T^{2} \) |
| 41 | \( 1 + (55.7 - 26.8i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (148. + 71.4i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-337. - 423. i)T + (-2.31e4 + 1.01e5i)T^{2} \) |
| 53 | \( 1 + (108. - 474. i)T + (-1.34e5 - 6.45e4i)T^{2} \) |
| 59 | \( 1 + (-203. - 97.8i)T + (1.28e5 + 1.60e5i)T^{2} \) |
| 61 | \( 1 + (-103. - 452. i)T + (-2.04e5 + 9.84e4i)T^{2} \) |
| 67 | \( 1 + 133.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-156. + 685. i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-704. + 883. i)T + (-8.65e4 - 3.79e5i)T^{2} \) |
| 79 | \( 1 + 1.18e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-673. + 844. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (881. - 1.10e3i)T + (-1.56e5 - 6.87e5i)T^{2} \) |
| 97 | \( 1 + 352.T + 9.12e5T^{2} \) |
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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
−15.20878666574691167545030243049, −14.05304052376012288582472066457, −13.34351120230456108449056044669, −12.66011908527676008492786273254, −10.45349812141151578874656759899, −8.851438037157692156668021053962, −7.71155741834810482526397603972, −6.06353803158746223798107563176, −5.30261043193642085257673757314, −2.95758172906304421919411225249,
2.37717457061510321269828120476, 3.58236182351110142419641551377, 5.33213066210638688134445691770, 7.21772779539984435439787012463, 9.637478907199566770456832745730, 10.01011535423996537286114896778, 11.55096427150294595595070434112, 12.76338124135992421949904840091, 13.67282119756522091067069425431, 14.39936623631263591146699052277
