| L(s) = 1 | + (0.467 − 1.28i)3-s + (1.73 − 0.305i)5-s + (−1.91 + 3.31i)7-s + (0.868 + 0.728i)9-s + (2.26 − 1.30i)11-s + (2.03 + 5.60i)13-s + (0.417 − 2.36i)15-s + (2.13 − 1.78i)17-s + (−0.713 + 4.30i)19-s + (3.36 + 4.00i)21-s + (1.10 − 6.27i)23-s + (−1.78 + 0.650i)25-s + (4.89 − 2.82i)27-s + (2.78 − 3.31i)29-s + (−2.41 + 4.18i)31-s + ⋯ |
| L(s) = 1 | + (0.269 − 0.741i)3-s + (0.775 − 0.136i)5-s + (−0.723 + 1.25i)7-s + (0.289 + 0.242i)9-s + (0.682 − 0.394i)11-s + (0.565 + 1.55i)13-s + (0.107 − 0.611i)15-s + (0.516 − 0.433i)17-s + (−0.163 + 0.986i)19-s + (0.733 + 0.873i)21-s + (0.230 − 1.30i)23-s + (−0.357 + 0.130i)25-s + (0.941 − 0.543i)27-s + (0.516 − 0.615i)29-s + (−0.433 + 0.751i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0383i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.85355 + 0.0355500i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.85355 + 0.0355500i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (0.713 - 4.30i)T \) |
| good | 3 | \( 1 + (-0.467 + 1.28i)T + (-2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-1.73 + 0.305i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (1.91 - 3.31i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.26 + 1.30i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.03 - 5.60i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.13 + 1.78i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-1.10 + 6.27i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.78 + 3.31i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (2.41 - 4.18i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7.50iT - 37T^{2} \) |
| 41 | \( 1 + (6.78 + 2.46i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-3.24 + 0.571i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (3.73 + 3.13i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (1.49 + 0.264i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-4.81 - 5.74i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (5.64 + 0.995i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.11 + 2.51i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (0.236 + 1.34i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-10.5 - 3.83i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-6.51 - 2.37i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-5.89 - 3.40i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (6.91 - 2.51i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (9.33 - 7.83i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.56097741721647597329101227121, −9.524087217122482556026267449574, −8.982256780625914401696412652847, −8.170942062612025659561664743914, −6.78283968233124650007150998834, −6.33211178600282157680475997018, −5.37346469554205283789108979933, −3.91634703537960890848474097143, −2.47198735577228548716324821541, −1.60621375120401434171730072160,
1.19244251688227842013860007221, 3.17975842180787550727962625143, 3.80781669425754398586915207427, 5.00066192659907532106097801372, 6.19479858825696052477681870604, 6.96933056456489475924835527325, 8.014518412098663954789449415786, 9.259251902671235683913494775648, 9.883957287559254225031825606771, 10.32128389995216536242928382796
