L(s) = 1 | + (−1.43 − 1.76i)2-s + (−1.19 + 1.03i)3-s + (−0.646 + 3.08i)4-s + (0.511 + 1.61i)5-s + (3.52 + 0.624i)6-s + (−1.13 + 3.29i)7-s + (2.32 − 1.19i)8-s + (−0.0729 + 0.503i)9-s + (2.11 − 3.21i)10-s + (−4.01 + 1.29i)11-s + (−2.41 − 4.34i)12-s + (2.63 − 0.890i)13-s + (7.42 − 2.71i)14-s + (−2.28 − 1.40i)15-s + (0.343 + 0.150i)16-s + (−4.32 − 2.15i)17-s + ⋯ |
L(s) = 1 | + (−1.01 − 1.24i)2-s + (−0.689 + 0.596i)3-s + (−0.323 + 1.54i)4-s + (0.228 + 0.723i)5-s + (1.44 + 0.254i)6-s + (−0.428 + 1.24i)7-s + (0.820 − 0.422i)8-s + (−0.0243 + 0.167i)9-s + (0.669 − 1.01i)10-s + (−1.21 + 0.391i)11-s + (−0.696 − 1.25i)12-s + (0.731 − 0.246i)13-s + (1.98 − 0.725i)14-s + (−0.588 − 0.361i)15-s + (0.0858 + 0.0376i)16-s + (−1.04 − 0.523i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.143i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00475627 + 0.0660099i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00475627 + 0.0660099i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 503 | \( 1 + (19.9 + 10.3i)T \) |
good | 2 | \( 1 + (1.43 + 1.76i)T + (-0.410 + 1.95i)T^{2} \) |
| 3 | \( 1 + (1.19 - 1.03i)T + (0.430 - 2.96i)T^{2} \) |
| 5 | \( 1 + (-0.511 - 1.61i)T + (-4.08 + 2.87i)T^{2} \) |
| 7 | \( 1 + (1.13 - 3.29i)T + (-5.51 - 4.31i)T^{2} \) |
| 11 | \( 1 + (4.01 - 1.29i)T + (8.91 - 6.44i)T^{2} \) |
| 13 | \( 1 + (-2.63 + 0.890i)T + (10.3 - 7.87i)T^{2} \) |
| 17 | \( 1 + (4.32 + 2.15i)T + (10.2 + 13.5i)T^{2} \) |
| 19 | \( 1 + (2.53 + 5.67i)T + (-12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (-0.991 + 3.42i)T + (-19.4 - 12.2i)T^{2} \) |
| 29 | \( 1 + (0.577 - 4.83i)T + (-28.1 - 6.83i)T^{2} \) |
| 31 | \( 1 + (0.390 + 4.79i)T + (-30.5 + 5.02i)T^{2} \) |
| 37 | \( 1 + (-2.72 - 3.44i)T + (-8.49 + 36.0i)T^{2} \) |
| 41 | \( 1 + (2.36 - 12.0i)T + (-37.9 - 15.5i)T^{2} \) |
| 43 | \( 1 + (-1.51 + 8.24i)T + (-40.1 - 15.2i)T^{2} \) |
| 47 | \( 1 + (0.677 - 0.0254i)T + (46.8 - 3.52i)T^{2} \) |
| 53 | \( 1 + (0.739 - 1.65i)T + (-35.4 - 39.4i)T^{2} \) |
| 59 | \( 1 + (-3.36 + 8.08i)T + (-41.5 - 41.8i)T^{2} \) |
| 61 | \( 1 + (-3.38 - 5.58i)T + (-28.2 + 54.0i)T^{2} \) |
| 67 | \( 1 + (5.05 - 3.10i)T + (30.3 - 59.7i)T^{2} \) |
| 71 | \( 1 + (-1.24 - 7.89i)T + (-67.5 + 21.8i)T^{2} \) |
| 73 | \( 1 + (2.58 + 2.47i)T + (3.19 + 72.9i)T^{2} \) |
| 79 | \( 1 + (-0.643 - 11.4i)T + (-78.4 + 8.88i)T^{2} \) |
| 83 | \( 1 + (8.03 + 15.3i)T + (-47.3 + 68.1i)T^{2} \) |
| 89 | \( 1 + (3.25 + 12.4i)T + (-77.5 + 43.6i)T^{2} \) |
| 97 | \( 1 + (-0.373 - 1.58i)T + (-86.7 + 43.3i)T^{2} \) |
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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
−11.08191749666417860197775346589, −10.57415315757685617716709172684, −9.853190219479765934192084183223, −8.953560390614206280819273350181, −8.249230344985123471028913925445, −6.78333837669144969133104361906, −5.69481926489004799493757736232, −4.61192048330478516462088424162, −2.80864198064730814602701107474, −2.41488051565649595285974993184,
0.06362946723522772024435117425, 1.28406536882520552516884252884, 3.85152811244353622237080882525, 5.38480994199748087776332642521, 6.13169353321075883530692272950, 6.85935437317173872258138297364, 7.71632043580988299054204150219, 8.519097843365414855524794889867, 9.368318288362531053635458345377, 10.40517300271680954959615661390