| L(s) = 1 | + (−0.846 + 3.71i)2-s + (3.04 − 3.82i)3-s + (−5.84 − 2.81i)4-s + (−12.1 + 15.1i)5-s + (11.6 + 14.5i)6-s + (13.1 + 13.0i)7-s + (−3.59 + 4.51i)8-s + (0.689 + 3.02i)9-s + (−46.0 − 57.7i)10-s + (3.84 − 16.8i)11-s + (−28.5 + 13.7i)12-s + (6.63 − 29.0i)13-s + (−59.4 + 37.8i)14-s + (21.1 + 92.5i)15-s + (−46.0 − 57.7i)16-s + (30.1 − 14.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.299 + 1.31i)2-s + (0.586 − 0.735i)3-s + (−0.730 − 0.351i)4-s + (−1.08 + 1.35i)5-s + (0.789 + 0.989i)6-s + (0.711 + 0.703i)7-s + (−0.159 + 0.199i)8-s + (0.0255 + 0.111i)9-s + (−1.45 − 1.82i)10-s + (0.105 − 0.462i)11-s + (−0.686 + 0.330i)12-s + (0.141 − 0.620i)13-s + (−1.13 + 0.722i)14-s + (0.363 + 1.59i)15-s + (−0.719 − 0.901i)16-s + (0.429 − 0.206i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.656 - 0.753i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.656 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.506240 + 1.11257i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.506240 + 1.11257i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-13.1 - 13.0i)T \) |
| good | 2 | \( 1 + (0.846 - 3.71i)T + (-7.20 - 3.47i)T^{2} \) |
| 3 | \( 1 + (-3.04 + 3.82i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (12.1 - 15.1i)T + (-27.8 - 121. i)T^{2} \) |
| 11 | \( 1 + (-3.84 + 16.8i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (-6.63 + 29.0i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 + (-30.1 + 14.4i)T + (3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 - 122.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-35.7 - 17.2i)T + (7.58e3 + 9.51e3i)T^{2} \) |
| 29 | \( 1 + (257. - 123. i)T + (1.52e4 - 1.90e4i)T^{2} \) |
| 31 | \( 1 - 61.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-34.5 + 16.6i)T + (3.15e4 - 3.96e4i)T^{2} \) |
| 41 | \( 1 + (-312. + 392. i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 43 | \( 1 + (51.8 + 65.0i)T + (-1.76e4 + 7.75e4i)T^{2} \) |
| 47 | \( 1 + (5.83 - 25.5i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (-386. - 186. i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + (-106. - 133. i)T + (-4.57e4 + 2.00e5i)T^{2} \) |
| 61 | \( 1 + (542. - 261. i)T + (1.41e5 - 1.77e5i)T^{2} \) |
| 67 | \( 1 + 606.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (278. + 134. i)T + (2.23e5 + 2.79e5i)T^{2} \) |
| 73 | \( 1 + (-0.148 - 0.649i)T + (-3.50e5 + 1.68e5i)T^{2} \) |
| 79 | \( 1 - 50.9T + 4.93e5T^{2} \) |
| 83 | \( 1 + (221. + 971. i)T + (-5.15e5 + 2.48e5i)T^{2} \) |
| 89 | \( 1 + (-225. - 990. i)T + (-6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 + 66.6T + 9.12e5T^{2} \) |
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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
−15.36963119365509025879094473013, −14.65680763014012542218295647289, −13.85387488778937149830029465703, −11.98923494494672745489457433986, −10.96061712775719592463977194760, −8.798731871972643702974575570786, −7.58799083354713669382174051099, −7.37925894557135344388439476074, −5.64326812601593093925728726923, −2.95864251067986166921099800424,
1.10804864689591873558878203515, 3.68104497269954436237226583695, 4.55605938806176445449520601409, 7.73876752623476622145641675936, 9.018623589136625920129435702164, 9.809077497695486209314999676794, 11.32730063059975394581687665645, 12.01233837969439479775345053304, 13.20841247423239901578917383912, 14.76512167571327788312277657428
