| L(s) = 1 | + (1.15 + 1.65i)2-s + (−0.352 + 1.69i)3-s + (−0.704 + 1.93i)4-s + (−2.13 − 0.650i)5-s + (−3.20 + 1.37i)6-s + (1.32 + 0.618i)7-s + (−0.115 + 0.0310i)8-s + (−2.75 − 1.19i)9-s + (−1.39 − 4.28i)10-s + (1.86 − 2.21i)11-s + (−3.03 − 1.87i)12-s + (5.02 + 3.51i)13-s + (0.512 + 2.90i)14-s + (1.85 − 3.39i)15-s + (2.97 + 2.49i)16-s + (−3.10 − 0.833i)17-s + ⋯ |
| L(s) = 1 | + (0.817 + 1.16i)2-s + (−0.203 + 0.979i)3-s + (−0.352 + 0.967i)4-s + (−0.956 − 0.290i)5-s + (−1.30 + 0.562i)6-s + (0.501 + 0.233i)7-s + (−0.0409 + 0.0109i)8-s + (−0.916 − 0.398i)9-s + (−0.442 − 1.35i)10-s + (0.560 − 0.668i)11-s + (−0.875 − 0.541i)12-s + (1.39 + 0.975i)13-s + (0.136 + 0.776i)14-s + (0.479 − 0.877i)15-s + (0.742 + 0.623i)16-s + (−0.753 − 0.202i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.590 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.640918 + 1.26352i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.640918 + 1.26352i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.352 - 1.69i)T \) |
| 5 | \( 1 + (2.13 + 0.650i)T \) |
| good | 2 | \( 1 + (-1.15 - 1.65i)T + (-0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (-1.32 - 0.618i)T + (4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (-1.86 + 2.21i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-5.02 - 3.51i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (3.10 + 0.833i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (3.51 + 2.03i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.14 + 4.58i)T + (-14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (0.368 - 2.08i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (3.20 + 1.16i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-3.04 + 11.3i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.839 + 0.147i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (0.732 - 0.0641i)T + (42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (1.21 - 2.61i)T + (-30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (-0.0757 - 0.0757i)T + 53iT^{2} \) |
| 59 | \( 1 + (-2.89 + 2.43i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (4.21 - 1.53i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (2.55 - 3.65i)T + (-22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (7.84 - 4.53i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.15 - 8.04i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (1.44 + 0.254i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (13.8 - 9.69i)T + (28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (3.05 - 5.28i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.41 - 16.1i)T + (-95.5 + 16.8i)T^{2} \) |
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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
−14.03042051194205192040910647980, −12.77786428053402341367718598119, −11.39546815354242346027328048519, −10.93112318296558055754907837226, −8.923181465438258710492168784754, −8.395125834264113587539014520227, −6.79799753492759354879684530170, −5.79088202241553883422193702424, −4.45606850330839109950739034462, −3.90177051267391157828949090043,
1.58951595793557411080764668112, 3.32485912800040538029223126542, 4.50541214830278060756566261022, 6.14084486670982419263559681614, 7.54139886732101487581712061624, 8.421266568294924688572107730409, 10.44793074538716076520093082775, 11.26353929655116400882618733149, 11.82366328742190887344635438899, 12.80427016309095432089384947003
