L(s) = 1 | + (1.28 + 1.28i)2-s + (2.65 − 1.53i)3-s + 1.30i·4-s + (−1.07 − 0.288i)5-s + (5.38 + 1.44i)6-s + (0.899 − 0.899i)8-s + (3.21 − 5.56i)9-s + (−1.01 − 1.75i)10-s + (0.124 + 0.0332i)11-s + (1.99 + 3.45i)12-s + (−3.59 + 0.291i)13-s + (−3.30 + 0.884i)15-s + 4.90·16-s + 0.273·17-s + (11.2 − 3.02i)18-s + (1.02 + 3.83i)19-s + ⋯ |
L(s) = 1 | + (0.908 + 0.908i)2-s + (1.53 − 0.886i)3-s + 0.650i·4-s + (−0.480 − 0.128i)5-s + (2.19 + 0.589i)6-s + (0.317 − 0.317i)8-s + (1.07 − 1.85i)9-s + (−0.319 − 0.553i)10-s + (0.0374 + 0.0100i)11-s + (0.576 + 0.997i)12-s + (−0.996 + 0.0807i)13-s + (−0.852 + 0.228i)15-s + 1.22·16-s + 0.0663·17-s + (2.65 − 0.711i)18-s + (0.235 + 0.879i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0432i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.56926 + 0.0772760i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.56926 + 0.0772760i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (3.59 - 0.291i)T \) |
good | 2 | \( 1 + (-1.28 - 1.28i)T + 2iT^{2} \) |
| 3 | \( 1 + (-2.65 + 1.53i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.07 + 0.288i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.124 - 0.0332i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 - 0.273T + 17T^{2} \) |
| 19 | \( 1 + (-1.02 - 3.83i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + 0.491iT - 23T^{2} \) |
| 29 | \( 1 + (4.62 - 8.00i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.85 - 6.92i)T + (-26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (2.82 - 2.82i)T - 37iT^{2} \) |
| 41 | \( 1 + (-2.31 - 8.63i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-8.44 + 4.87i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.51 + 5.66i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.467 + 0.809i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.86 + 4.86i)T + 59iT^{2} \) |
| 61 | \( 1 + (6.74 + 3.89i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.316 - 1.18i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (0.793 - 2.96i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (0.118 - 0.0318i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (5.72 + 9.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.07 + 8.07i)T - 83iT^{2} \) |
| 89 | \( 1 + (2.99 + 2.99i)T + 89iT^{2} \) |
| 97 | \( 1 + (2.39 + 0.641i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.39196028207788681850775355036, −9.432201986513988598100909147985, −8.480844954977569467968989425082, −7.64759242770104196653605823871, −7.23355973996004037591496021073, −6.30966338126448223700143545603, −5.05117445425975898066579888636, −3.94090731165345067736006973484, −3.07733611755958460729481178730, −1.60645932166581575986381109138,
2.24425296373830769223549522906, 2.86442533360242888952055206854, 3.97275071603986584130652186667, 4.36057054191802402849510411771, 5.53166459247862916937231361037, 7.54862671715638158047511113634, 7.83040298499949228610543682184, 9.145980103358099032993303854987, 9.691365561785637135733597178333, 10.65321570118351170239194773805