| L(s) = 1 | + (0.997 + 0.0697i)2-s + (−0.0568 − 1.62i)3-s + (0.990 + 0.139i)4-s + (0.280 − 2.21i)5-s + (0.0568 − 1.62i)6-s + (−1.45 − 2.52i)7-s + (0.978 + 0.207i)8-s + (0.348 − 0.0243i)9-s + (0.434 − 2.19i)10-s + (−0.0569 − 0.541i)11-s + (0.170 − 1.61i)12-s + (−1.45 − 2.15i)13-s + (−1.27 − 2.61i)14-s + (−3.62 − 0.329i)15-s + (0.961 + 0.275i)16-s + (−1.13 + 2.80i)17-s + ⋯ |
| L(s) = 1 | + (0.705 + 0.0493i)2-s + (−0.0328 − 0.939i)3-s + (0.495 + 0.0695i)4-s + (0.125 − 0.992i)5-s + (0.0231 − 0.664i)6-s + (−0.550 − 0.953i)7-s + (0.345 + 0.0735i)8-s + (0.116 − 0.00811i)9-s + (0.137 − 0.693i)10-s + (−0.0171 − 0.163i)11-s + (0.0491 − 0.467i)12-s + (−0.404 − 0.599i)13-s + (−0.341 − 0.699i)14-s + (−0.936 − 0.0851i)15-s + (0.240 + 0.0689i)16-s + (−0.275 + 0.681i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 + 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.628 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.954330 - 1.99715i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.954330 - 1.99715i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.997 - 0.0697i)T \) |
| 5 | \( 1 + (-0.280 + 2.21i)T \) |
| 19 | \( 1 + (-2.00 - 3.86i)T \) |
| good | 3 | \( 1 + (0.0568 + 1.62i)T + (-2.99 + 0.209i)T^{2} \) |
| 7 | \( 1 + (1.45 + 2.52i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.0569 + 0.541i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (1.45 + 2.15i)T + (-4.86 + 12.0i)T^{2} \) |
| 17 | \( 1 + (1.13 - 2.80i)T + (-12.2 - 11.8i)T^{2} \) |
| 23 | \( 1 + (-1.42 - 1.38i)T + (0.802 + 22.9i)T^{2} \) |
| 29 | \( 1 + (1.43 + 3.56i)T + (-20.8 + 20.1i)T^{2} \) |
| 31 | \( 1 + (4.35 - 4.84i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (-3.83 - 2.78i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-2.12 - 0.608i)T + (34.7 + 21.7i)T^{2} \) |
| 43 | \( 1 + (-1.93 + 10.9i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.875 - 2.16i)T + (-33.8 + 32.6i)T^{2} \) |
| 53 | \( 1 + (-6.66 - 0.936i)T + (50.9 + 14.6i)T^{2} \) |
| 59 | \( 1 + (1.19 + 4.77i)T + (-52.0 + 27.6i)T^{2} \) |
| 61 | \( 1 + (-7.77 - 7.50i)T + (2.12 + 60.9i)T^{2} \) |
| 67 | \( 1 + (10.2 + 6.42i)T + (29.3 + 60.2i)T^{2} \) |
| 71 | \( 1 + (0.393 + 0.209i)T + (39.7 + 58.8i)T^{2} \) |
| 73 | \( 1 + (2.47 - 3.67i)T + (-27.3 - 67.6i)T^{2} \) |
| 79 | \( 1 + (0.0542 + 1.55i)T + (-78.8 + 5.51i)T^{2} \) |
| 83 | \( 1 + (-6.89 + 7.65i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-0.267 + 0.0766i)T + (75.4 - 47.1i)T^{2} \) |
| 97 | \( 1 + (-11.2 + 7.04i)T + (42.5 - 87.1i)T^{2} \) |
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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
−9.916835629262677760727538651055, −8.752948571838452399844660638486, −7.72317200842160238227178356999, −7.26352466214130561531946323031, −6.22901277716162573329711688789, −5.48627122810835272345650697108, −4.34994563257245998286162537757, −3.51092633627609398072747199126, −1.95986044772158361811306467640, −0.827857536226164219019151187783,
2.30454077492577832790786029435, 3.06197902583132741354183641785, 4.11276047959789783051833010196, 5.01700400572155626254087745291, 5.89902540384226154119174144966, 6.84314054913518710357087813887, 7.48494587844464704951519848801, 9.173771561880204259755639504416, 9.461253538375820138932738171057, 10.39596287049908101333732421799
