| L(s) = 1 | + (0.231 + 0.903i)2-s + (0.325 − 0.0620i)3-s + (0.990 − 0.544i)4-s + (0.131 + 0.279i)6-s + (−2.39 + 3.30i)7-s + (1.99 + 2.12i)8-s + (−2.68 + 1.06i)9-s + (−5.98 + 1.53i)11-s + (0.288 − 0.238i)12-s + (0.617 + 1.56i)13-s + (−3.53 − 1.40i)14-s + (−0.246 + 0.388i)16-s + (−1.35 + 2.47i)17-s + (−1.58 − 2.18i)18-s + (0.447 − 2.34i)19-s + ⋯ |
| L(s) = 1 | + (0.163 + 0.638i)2-s + (0.187 − 0.0358i)3-s + (0.495 − 0.272i)4-s + (0.0537 + 0.114i)6-s + (−0.907 + 1.24i)7-s + (0.706 + 0.752i)8-s + (−0.895 + 0.354i)9-s + (−1.80 + 0.463i)11-s + (0.0833 − 0.0689i)12-s + (0.171 + 0.432i)13-s + (−0.945 − 0.374i)14-s + (−0.0616 + 0.0971i)16-s + (−0.329 + 0.599i)17-s + (−0.373 − 0.513i)18-s + (0.102 − 0.538i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 - 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.476187 + 1.23342i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.476187 + 1.23342i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + (-0.231 - 0.903i)T + (-1.75 + 0.963i)T^{2} \) |
| 3 | \( 1 + (-0.325 + 0.0620i)T + (2.78 - 1.10i)T^{2} \) |
| 7 | \( 1 + (2.39 - 3.30i)T + (-2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (5.98 - 1.53i)T + (9.63 - 5.29i)T^{2} \) |
| 13 | \( 1 + (-0.617 - 1.56i)T + (-9.47 + 8.89i)T^{2} \) |
| 17 | \( 1 + (1.35 - 2.47i)T + (-9.10 - 14.3i)T^{2} \) |
| 19 | \( 1 + (-0.447 + 2.34i)T + (-17.6 - 6.99i)T^{2} \) |
| 23 | \( 1 + (-3.56 - 0.224i)T + (22.8 + 2.88i)T^{2} \) |
| 29 | \( 1 + (-2.50 - 0.316i)T + (28.0 + 7.21i)T^{2} \) |
| 31 | \( 1 + (-8.14 - 4.47i)T + (16.6 + 26.1i)T^{2} \) |
| 37 | \( 1 + (0.404 + 0.256i)T + (15.7 + 33.4i)T^{2} \) |
| 41 | \( 1 + (0.485 + 7.71i)T + (-40.6 + 5.13i)T^{2} \) |
| 43 | \( 1 + (-5.98 + 1.94i)T + (34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (3.47 - 3.70i)T + (-2.95 - 46.9i)T^{2} \) |
| 53 | \( 1 + (-5.70 - 2.68i)T + (33.7 + 40.8i)T^{2} \) |
| 59 | \( 1 + (-2.17 - 2.62i)T + (-11.0 + 57.9i)T^{2} \) |
| 61 | \( 1 + (-0.476 + 7.57i)T + (-60.5 - 7.64i)T^{2} \) |
| 67 | \( 1 + (-0.591 - 4.68i)T + (-64.8 + 16.6i)T^{2} \) |
| 71 | \( 1 + (4.72 + 4.43i)T + (4.45 + 70.8i)T^{2} \) |
| 73 | \( 1 + (5.24 + 4.33i)T + (13.6 + 71.7i)T^{2} \) |
| 79 | \( 1 + (-2.35 - 12.3i)T + (-73.4 + 29.0i)T^{2} \) |
| 83 | \( 1 + (17.6 + 3.37i)T + (77.1 + 30.5i)T^{2} \) |
| 89 | \( 1 + (0.523 - 0.633i)T + (-16.6 - 87.4i)T^{2} \) |
| 97 | \( 1 + (0.164 - 1.30i)T + (-93.9 - 24.1i)T^{2} \) |
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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
−10.84393560156761151381515906870, −10.16865812584234914771226360437, −8.948339358824601398460175162927, −8.311559347107056466107143185784, −7.31296542854907009086288083284, −6.36128353462502762948587691897, −5.58387849668002368589029971003, −4.90205477431379651651672329233, −2.85632712152768809654166259285, −2.34721180306674366363424470986,
0.61695624952762124745125938827, 2.78074985136052647798893599171, 3.12511590616397286106795543184, 4.36856063301703831142588200049, 5.75805250944481465257705296187, 6.76478978073316322546987530109, 7.66477071104317218436959699157, 8.387906448497095006929044945208, 9.832570819067209297349491202925, 10.33718022131330204659778200135
