| L(s) = 1 | + (−0.951 + 0.309i)2-s + (−0.454 + 0.330i)3-s + (0.809 − 0.587i)4-s + (2.02 − 0.946i)5-s + (0.330 − 0.454i)6-s + 3.59i·7-s + (−0.587 + 0.809i)8-s + (−0.829 + 2.55i)9-s + (−1.63 + 1.52i)10-s + (−3.90 + 1.26i)11-s + (−0.173 + 0.534i)12-s + (0.643 − 3.54i)13-s + (−1.10 − 3.41i)14-s + (−0.608 + 1.09i)15-s + (0.309 − 0.951i)16-s + (−0.477 − 0.346i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (−0.262 + 0.190i)3-s + (0.404 − 0.293i)4-s + (0.905 − 0.423i)5-s + (0.134 − 0.185i)6-s + 1.35i·7-s + (−0.207 + 0.286i)8-s + (−0.276 + 0.850i)9-s + (−0.516 + 0.482i)10-s + (−1.17 + 0.382i)11-s + (−0.0501 + 0.154i)12-s + (0.178 − 0.983i)13-s + (−0.296 − 0.912i)14-s + (−0.157 + 0.283i)15-s + (0.0772 − 0.237i)16-s + (−0.115 − 0.0840i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.292543 + 0.679644i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.292543 + 0.679644i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 5 | \( 1 + (-2.02 + 0.946i)T \) |
| 13 | \( 1 + (-0.643 + 3.54i)T \) |
| good | 3 | \( 1 + (0.454 - 0.330i)T + (0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 - 3.59iT - 7T^{2} \) |
| 11 | \( 1 + (3.90 - 1.26i)T + (8.89 - 6.46i)T^{2} \) |
| 17 | \( 1 + (0.477 + 0.346i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (4.56 - 6.28i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.238 - 0.734i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (2.30 - 1.67i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.23 - 1.70i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-6.46 - 2.10i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (7.00 + 2.27i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 2.01T + 43T^{2} \) |
| 47 | \( 1 + (-2.01 - 2.78i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (6.24 - 4.53i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (4.82 + 1.56i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-4.02 - 12.3i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-3.99 + 5.50i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-3.16 - 4.34i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (15.0 - 4.88i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (9.97 - 7.24i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-3.41 + 4.69i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-11.9 + 3.89i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-9.53 - 13.1i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−10.40128824679756507232447525967, −10.22733354203258266133974741305, −9.042625440511987741136106853319, −8.362913031466337869607457443653, −7.66838877613914457164558744486, −6.05565722254292787093998807085, −5.64476668393036376924180597442, −4.87332998717801304338992642746, −2.73490227629115554171153473897, −1.89800285172946693019406908493,
0.47930725110613376081491900685, 2.06504320518817118951835879467, 3.30604607976020301077632510075, 4.61624220462999549087593428895, 6.06230548843530159096618957679, 6.73348973584994615062226533477, 7.46365728382927806535958176324, 8.671869376390767654799501731413, 9.458999634985353058771141478813, 10.29216360028509910272897458331
