| L(s) = 1 | + (0.673 − 0.565i)2-s + (−0.5 − 0.181i)3-s + (−0.213 + 1.20i)4-s + (−0.439 − 2.49i)5-s + (−0.439 + 0.160i)6-s + (0.939 − 1.62i)7-s + (1.41 + 2.45i)8-s + (−2.08 − 1.74i)9-s + (−1.70 − 1.43i)10-s + (−1.70 − 2.95i)11-s + (0.326 − 0.565i)12-s + (4.97 − 1.80i)13-s + (−0.286 − 1.62i)14-s + (−0.233 + 1.32i)15-s + (0.0393 + 0.0143i)16-s + (1.26 − 1.06i)17-s + ⋯ |
| L(s) = 1 | + (0.476 − 0.399i)2-s + (−0.288 − 0.105i)3-s + (−0.106 + 0.604i)4-s + (−0.196 − 1.11i)5-s + (−0.179 + 0.0653i)6-s + (0.355 − 0.615i)7-s + (0.501 + 0.868i)8-s + (−0.693 − 0.582i)9-s + (−0.539 − 0.452i)10-s + (−0.514 − 0.890i)11-s + (0.0942 − 0.163i)12-s + (1.37 − 0.501i)13-s + (−0.0767 − 0.434i)14-s + (−0.0604 + 0.342i)15-s + (0.00984 + 0.00358i)16-s + (0.307 − 0.257i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0225 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0225 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.987157 - 1.00963i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.987157 - 1.00963i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 + (-0.673 + 0.565i)T + (0.347 - 1.96i)T^{2} \) |
| 3 | \( 1 + (0.5 + 0.181i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (0.439 + 2.49i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.939 + 1.62i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.70 + 2.95i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.97 + 1.80i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.26 + 1.06i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.305 + 1.73i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (2.65 + 2.22i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.971 - 1.68i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 0.837T + 37T^{2} \) |
| 41 | \( 1 + (-4.21 - 1.53i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.833 - 4.72i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.549 - 0.460i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (1.06 - 6.01i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (8.24 - 6.91i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.762 + 4.32i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-10.8 - 9.13i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (2.38 + 13.5i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-7.06 - 2.57i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-6.54 - 2.38i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (1.25 - 2.17i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.14 + 0.780i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (1.39 - 1.17i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.24876223299071857594521667611, −10.79855117710387568672301609673, −9.085895161409738623031912015366, −8.396191234276323721058383791862, −7.72219335655283399089749603715, −6.08360934708663124568910983060, −5.14838189516776108163485804116, −4.05630584163645643977893881541, −3.07803749939997106763225021012, −0.924617003860465149584321876849,
2.06935158739278884231197932710, 3.66769800722181307994867535455, 5.01395742213913391112274959464, 5.80953300886696920783164711024, 6.68599221918500012450531808637, 7.69621500876786051953017363383, 8.885853435457797341978367044044, 10.06700965075297052847425742684, 10.90486427502737144171648592468, 11.35838743204437542404361887286
