| L(s) = 1 | + (−0.109 − 1.45i)3-s + (1.29 − 0.398i)5-s + (0.108 − 0.187i)7-s + (0.860 − 0.129i)9-s + (3.76 + 4.71i)11-s + (2.10 + 1.95i)13-s + (−0.720 − 1.83i)15-s + (0.270 + 0.0833i)17-s + (−1.12 − 0.169i)19-s + (−0.284 − 0.136i)21-s + (1.44 − 3.67i)23-s + (−2.62 + 1.78i)25-s + (−1.25 − 5.50i)27-s + (0.515 − 6.88i)29-s + (8.17 + 5.57i)31-s + ⋯ |
| L(s) = 1 | + (−0.0629 − 0.840i)3-s + (0.577 − 0.178i)5-s + (0.0408 − 0.0708i)7-s + (0.286 − 0.0432i)9-s + (1.13 + 1.42i)11-s + (0.584 + 0.542i)13-s + (−0.185 − 0.473i)15-s + (0.0654 + 0.0202i)17-s + (−0.257 − 0.0388i)19-s + (−0.0620 − 0.0298i)21-s + (0.300 − 0.765i)23-s + (−0.524 + 0.357i)25-s + (−0.241 − 1.05i)27-s + (0.0958 − 1.27i)29-s + (1.46 + 1.00i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.74293 - 0.586083i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.74293 - 0.586083i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + (5.90 + 2.85i)T \) |
| good | 3 | \( 1 + (0.109 + 1.45i)T + (-2.96 + 0.447i)T^{2} \) |
| 5 | \( 1 + (-1.29 + 0.398i)T + (4.13 - 2.81i)T^{2} \) |
| 7 | \( 1 + (-0.108 + 0.187i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.76 - 4.71i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-2.10 - 1.95i)T + (0.971 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.270 - 0.0833i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (1.12 + 0.169i)T + (18.1 + 5.60i)T^{2} \) |
| 23 | \( 1 + (-1.44 + 3.67i)T + (-16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (-0.515 + 6.88i)T + (-28.6 - 4.32i)T^{2} \) |
| 31 | \( 1 + (-8.17 - 5.57i)T + (11.3 + 28.8i)T^{2} \) |
| 37 | \( 1 + (3.77 + 6.53i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.62 - 2.22i)T + (25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (0.288 - 0.361i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (6.12 - 5.68i)T + (3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (1.85 + 8.14i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (-11.0 + 7.56i)T + (22.2 - 56.7i)T^{2} \) |
| 67 | \( 1 + (6.22 + 0.938i)T + (64.0 + 19.7i)T^{2} \) |
| 71 | \( 1 + (-0.540 - 1.37i)T + (-52.0 + 48.2i)T^{2} \) |
| 73 | \( 1 + (-0.601 - 0.557i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-3.07 + 5.33i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.538 - 7.18i)T + (-82.0 + 12.3i)T^{2} \) |
| 89 | \( 1 + (-0.331 - 4.42i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + (2.78 + 3.49i)T + (-21.5 + 94.5i)T^{2} \) |
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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
−10.18760901551373996698316159726, −9.582967752660793069833200412390, −8.682059038635235914643040799586, −7.62404500484137071152340732059, −6.67158322825009737316557169739, −6.33018541584541710622862065529, −4.83539961041826543757613687640, −3.95296194269534025577080139528, −2.15292336697830263735830789191, −1.33813706882961663145470746984,
1.36137073736729049662931502506, 3.15622626268228656898258629133, 3.93731320252183746461741619109, 5.15155686851448622999793940194, 6.03131957665186841751923976494, 6.82311226937883626554648401402, 8.246836510736668238364044284256, 8.902674323142413565581456319789, 9.848274336532183530319704454116, 10.40693778686219318254481284013
