| L(s) = 1 | + (−0.102 − 0.126i)2-s + (0.592 − 0.513i)3-s + (0.404 − 1.93i)4-s + (−0.990 − 3.12i)5-s + (−0.125 − 0.0222i)6-s + (1.03 − 3.01i)7-s + (−0.575 + 0.296i)8-s + (−0.342 + 2.36i)9-s + (−0.293 + 0.446i)10-s + (1.46 − 0.472i)11-s + (−0.751 − 1.35i)12-s + (−3.42 + 1.15i)13-s + (−0.487 + 0.178i)14-s + (−2.19 − 1.34i)15-s + (−3.51 − 1.54i)16-s + (5.57 + 2.78i)17-s + ⋯ |
| L(s) = 1 | + (−0.0726 − 0.0894i)2-s + (0.342 − 0.296i)3-s + (0.202 − 0.965i)4-s + (−0.442 − 1.39i)5-s + (−0.0513 − 0.00909i)6-s + (0.392 − 1.13i)7-s + (−0.203 + 0.104i)8-s + (−0.114 + 0.786i)9-s + (−0.0928 + 0.141i)10-s + (0.440 − 0.142i)11-s + (−0.216 − 0.390i)12-s + (−0.950 + 0.320i)13-s + (−0.130 + 0.0476i)14-s + (−0.565 − 0.347i)15-s + (−0.879 − 0.385i)16-s + (1.35 + 0.675i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.515794 - 1.33257i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.515794 - 1.33257i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 503 | \( 1 + (20.6 - 8.83i)T \) |
| good | 2 | \( 1 + (0.102 + 0.126i)T + (-0.410 + 1.95i)T^{2} \) |
| 3 | \( 1 + (-0.592 + 0.513i)T + (0.430 - 2.96i)T^{2} \) |
| 5 | \( 1 + (0.990 + 3.12i)T + (-4.08 + 2.87i)T^{2} \) |
| 7 | \( 1 + (-1.03 + 3.01i)T + (-5.51 - 4.31i)T^{2} \) |
| 11 | \( 1 + (-1.46 + 0.472i)T + (8.91 - 6.44i)T^{2} \) |
| 13 | \( 1 + (3.42 - 1.15i)T + (10.3 - 7.87i)T^{2} \) |
| 17 | \( 1 + (-5.57 - 2.78i)T + (10.2 + 13.5i)T^{2} \) |
| 19 | \( 1 + (-0.0566 - 0.127i)T + (-12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (1.57 - 5.45i)T + (-19.4 - 12.2i)T^{2} \) |
| 29 | \( 1 + (-1.13 + 9.53i)T + (-28.1 - 6.83i)T^{2} \) |
| 31 | \( 1 + (-0.368 - 4.52i)T + (-30.5 + 5.02i)T^{2} \) |
| 37 | \( 1 + (-0.0478 - 0.0604i)T + (-8.49 + 36.0i)T^{2} \) |
| 41 | \( 1 + (-0.653 + 3.32i)T + (-37.9 - 15.5i)T^{2} \) |
| 43 | \( 1 + (-1.20 + 6.58i)T + (-40.1 - 15.2i)T^{2} \) |
| 47 | \( 1 + (-6.01 + 0.225i)T + (46.8 - 3.52i)T^{2} \) |
| 53 | \( 1 + (-5.36 + 12.0i)T + (-35.4 - 39.4i)T^{2} \) |
| 59 | \( 1 + (-3.66 + 8.81i)T + (-41.5 - 41.8i)T^{2} \) |
| 61 | \( 1 + (-2.64 - 4.37i)T + (-28.2 + 54.0i)T^{2} \) |
| 67 | \( 1 + (10.0 - 6.15i)T + (30.3 - 59.7i)T^{2} \) |
| 71 | \( 1 + (1.17 + 7.44i)T + (-67.5 + 21.8i)T^{2} \) |
| 73 | \( 1 + (4.75 + 4.55i)T + (3.19 + 72.9i)T^{2} \) |
| 79 | \( 1 + (-0.594 - 10.5i)T + (-78.4 + 8.88i)T^{2} \) |
| 83 | \( 1 + (3.66 + 7.01i)T + (-47.3 + 68.1i)T^{2} \) |
| 89 | \( 1 + (-2.06 - 7.87i)T + (-77.5 + 43.6i)T^{2} \) |
| 97 | \( 1 + (1.82 + 7.73i)T + (-86.7 + 43.3i)T^{2} \) |
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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
−10.41816631994600234809140544737, −9.812911272184693043634825472225, −8.744601089028236425943461955303, −7.86071891466062464029770860774, −7.20122538263874861753805543756, −5.68044119198269617361499807108, −4.89639028059329260171125150059, −3.93344558757116560884882946300, −1.91383924126950071401391237996, −0.879687830561931790629592535387,
2.70750087663154816707158237956, 3.06379390295218441125753940319, 4.31653805183564902775062554634, 5.85335151062765402321031876713, 6.93081273096132143512988786299, 7.58706489744029694893345188180, 8.553650558090246143489335696499, 9.375816749404422584205244973396, 10.36671515515708530713610788205, 11.44764885181658924356132142194
