| L(s) = 1 | + (−1.17 + 0.783i)2-s + (0.418 + 2.10i)3-s + (0.772 − 1.84i)4-s + (−0.561 + 0.840i)5-s + (−2.14 − 2.15i)6-s + (−1.73 + 1.15i)7-s + (0.534 + 2.77i)8-s + (−1.48 + 0.615i)9-s + (0.00292 − 1.42i)10-s + (1.04 + 0.208i)11-s + (4.20 + 0.854i)12-s + (4.36 − 4.36i)13-s + (1.13 − 2.71i)14-s + (−2.00 − 0.830i)15-s + (−2.80 − 2.85i)16-s + (0.0297 + 4.12i)17-s + ⋯ |
| L(s) = 1 | + (−0.832 + 0.553i)2-s + (0.241 + 1.21i)3-s + (0.386 − 0.922i)4-s + (−0.251 + 0.375i)5-s + (−0.874 − 0.878i)6-s + (−0.654 + 0.437i)7-s + (0.189 + 0.981i)8-s + (−0.495 + 0.205i)9-s + (0.000924 − 0.452i)10-s + (0.316 + 0.0629i)11-s + (1.21 + 0.246i)12-s + (1.21 − 1.21i)13-s + (0.302 − 0.726i)14-s + (−0.517 − 0.214i)15-s + (−0.701 − 0.712i)16-s + (0.00721 + 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.208 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.414176 + 0.511542i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.414176 + 0.511542i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.17 - 0.783i)T \) |
| 17 | \( 1 + (-0.0297 - 4.12i)T \) |
| good | 3 | \( 1 + (-0.418 - 2.10i)T + (-2.77 + 1.14i)T^{2} \) |
| 5 | \( 1 + (0.561 - 0.840i)T + (-1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (1.73 - 1.15i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-1.04 - 0.208i)T + (10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (-4.36 + 4.36i)T - 13iT^{2} \) |
| 19 | \( 1 + (-2.55 + 6.15i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-0.773 + 3.88i)T + (-21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (3.11 + 2.08i)T + (11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + (5.38 - 1.07i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (5.03 - 1.00i)T + (34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (4.02 + 6.02i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.733 - 1.76i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (6.00 + 6.00i)T + 47iT^{2} \) |
| 53 | \( 1 + (1.43 + 0.596i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-4.31 + 1.78i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (-6.25 + 4.18i)T + (23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 - 0.616T + 67T^{2} \) |
| 71 | \( 1 + (-1.17 - 5.88i)T + (-65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-0.112 + 0.168i)T + (-27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (-11.4 - 2.28i)T + (72.9 + 30.2i)T^{2} \) |
| 83 | \( 1 + (11.4 + 4.74i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-1.64 - 1.64i)T + 89iT^{2} \) |
| 97 | \( 1 + (4.27 + 2.85i)T + (37.1 + 89.6i)T^{2} \) |
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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
−15.45524930886196422935343901238, −14.64694365065485469275380642770, −13.04569499888119015179241505781, −11.14929912775735995458347546911, −10.41781059821514479915403870007, −9.316968809637519818060354757915, −8.468879778078912080721721978460, −6.80983924193464592121680390834, −5.40133137770664202139271970548, −3.41719459908729245195903107124,
1.48041859237111937737862124793, 3.64278827306858801483755933933, 6.54978690346212886393173846109, 7.49060781219697873488434911754, 8.653018242717275247885692370943, 9.760494987797280435495279456183, 11.33119211651783728604936151265, 12.20666787653739054844756188594, 13.18854876455738676716351101456, 14.01474417166959973260947755043
