L(s) = 1 | + (0.469 − 2.05i)2-s + (−3.54 + 4.44i)3-s + (3.19 + 1.53i)4-s + (−7.88 + 9.88i)5-s + (7.48 + 9.38i)6-s + (3.32 + 18.2i)7-s + (15.1 − 19.0i)8-s + (−1.19 − 5.22i)9-s + (16.6 + 20.8i)10-s + (2.22 − 9.75i)11-s + (−18.1 + 8.75i)12-s + (−2.85 + 12.5i)13-s + (39.0 + 1.71i)14-s + (−16.0 − 70.1i)15-s + (−14.3 − 18.0i)16-s + (45.2 − 21.7i)17-s + ⋯ |
L(s) = 1 | + (0.166 − 0.727i)2-s + (−0.682 + 0.855i)3-s + (0.399 + 0.192i)4-s + (−0.704 + 0.884i)5-s + (0.509 + 0.638i)6-s + (0.179 + 0.983i)7-s + (0.671 − 0.841i)8-s + (−0.0442 − 0.193i)9-s + (0.525 + 0.659i)10-s + (0.0610 − 0.267i)11-s + (−0.437 + 0.210i)12-s + (−0.0609 + 0.267i)13-s + (0.745 + 0.0328i)14-s + (−0.275 − 1.20i)15-s + (−0.224 − 0.281i)16-s + (0.645 − 0.310i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 - 0.875i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.483 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.04418 + 0.616056i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04418 + 0.616056i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-3.32 - 18.2i)T \) |
good | 2 | \( 1 + (-0.469 + 2.05i)T + (-7.20 - 3.47i)T^{2} \) |
| 3 | \( 1 + (3.54 - 4.44i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (7.88 - 9.88i)T + (-27.8 - 121. i)T^{2} \) |
| 11 | \( 1 + (-2.22 + 9.75i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (2.85 - 12.5i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 + (-45.2 + 21.7i)T + (3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 + 42.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + (45.8 + 22.1i)T + (7.58e3 + 9.51e3i)T^{2} \) |
| 29 | \( 1 + (-159. + 76.8i)T + (1.52e4 - 1.90e4i)T^{2} \) |
| 31 | \( 1 - 261.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-79.3 + 38.2i)T + (3.15e4 - 3.96e4i)T^{2} \) |
| 41 | \( 1 + (-254. + 319. i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 43 | \( 1 + (-298. - 374. i)T + (-1.76e4 + 7.75e4i)T^{2} \) |
| 47 | \( 1 + (94.4 - 413. i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (303. + 146. i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + (400. + 502. i)T + (-4.57e4 + 2.00e5i)T^{2} \) |
| 61 | \( 1 + (-451. + 217. i)T + (1.41e5 - 1.77e5i)T^{2} \) |
| 67 | \( 1 + 1.02e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-717. - 345. i)T + (2.23e5 + 2.79e5i)T^{2} \) |
| 73 | \( 1 + (-66.0 - 289. i)T + (-3.50e5 + 1.68e5i)T^{2} \) |
| 79 | \( 1 + 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-113. - 495. i)T + (-5.15e5 + 2.48e5i)T^{2} \) |
| 89 | \( 1 + (199. + 874. i)T + (-6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 - 185.T + 9.12e5T^{2} \) |
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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
−15.64713967892351104504638838933, −14.32110986195333076965755900889, −12.46620983355619601807665211920, −11.53135151268915634357952637661, −10.94651391809547643438763899785, −9.825733095152971138711813427616, −7.86993129600860938280241207460, −6.22416168386628717640934170208, −4.38096518911325531139065936050, −2.79635163284564371020746507018,
1.04843766827590131206218842948, 4.56087610838537095695047722596, 6.11031925334697072922146877360, 7.27201967728604770859114887187, 8.174825224358584520691763255177, 10.38095654406606833425509550009, 11.68590818004761219543523698503, 12.52616815029047372678856212903, 13.79055529747615677732718700838, 15.07158209457717223918203074057