| L(s) = 1 | + (−1.41 − 0.0575i)2-s + (2.52 + 0.856i)3-s + (1.99 + 0.162i)4-s + (0.753 − 1.52i)5-s + (−3.51 − 1.35i)6-s + (−2.49 − 0.887i)7-s + (−2.80 − 0.344i)8-s + (3.25 + 2.49i)9-s + (−1.15 + 2.11i)10-s + (1.91 + 1.67i)11-s + (4.89 + 2.11i)12-s + (2.02 − 3.03i)13-s + (3.47 + 1.39i)14-s + (3.21 − 3.21i)15-s + (3.94 + 0.648i)16-s + (4.36 − 1.16i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.0406i)2-s + (1.45 + 0.494i)3-s + (0.996 + 0.0813i)4-s + (0.337 − 0.683i)5-s + (−1.43 − 0.553i)6-s + (−0.942 − 0.335i)7-s + (−0.992 − 0.121i)8-s + (1.08 + 0.832i)9-s + (−0.364 + 0.669i)10-s + (0.576 + 0.505i)11-s + (1.41 + 0.611i)12-s + (0.561 − 0.840i)13-s + (0.927 + 0.373i)14-s + (0.829 − 0.829i)15-s + (0.986 + 0.162i)16-s + (1.05 − 0.283i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.50919 - 0.0752681i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50919 - 0.0752681i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 + 0.0575i)T \) |
| 7 | \( 1 + (2.49 + 0.887i)T \) |
| good | 3 | \( 1 + (-2.52 - 0.856i)T + (2.38 + 1.82i)T^{2} \) |
| 5 | \( 1 + (-0.753 + 1.52i)T + (-3.04 - 3.96i)T^{2} \) |
| 11 | \( 1 + (-1.91 - 1.67i)T + (1.43 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-2.02 + 3.03i)T + (-4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (-4.36 + 1.16i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.337 - 0.0221i)T + (18.8 + 2.47i)T^{2} \) |
| 23 | \( 1 + (-3.85 - 2.95i)T + (5.95 + 22.2i)T^{2} \) |
| 29 | \( 1 + (-1.73 + 0.344i)T + (26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + (6.04 + 3.48i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.47 - 7.04i)T + (-22.5 - 29.3i)T^{2} \) |
| 41 | \( 1 + (-3.24 + 1.34i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (10.4 + 2.08i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (2.27 + 0.608i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.42 + 3.90i)T + (-6.91 - 52.5i)T^{2} \) |
| 59 | \( 1 + (-0.0739 - 1.12i)T + (-58.4 + 7.70i)T^{2} \) |
| 61 | \( 1 + (7.50 + 8.55i)T + (-7.96 + 60.4i)T^{2} \) |
| 67 | \( 1 + (1.06 - 3.14i)T + (-53.1 - 40.7i)T^{2} \) |
| 71 | \( 1 + (4.66 - 11.2i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (16.2 + 2.13i)T + (70.5 + 18.8i)T^{2} \) |
| 79 | \( 1 + (-12.9 - 3.48i)T + (68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (3.57 - 5.34i)T + (-31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (-1.32 - 10.0i)T + (-85.9 + 23.0i)T^{2} \) |
| 97 | \( 1 - 0.573T + 97T^{2} \) |
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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
−10.57827144671906513264124200658, −9.758590581953717559896748157473, −9.369998745441027466491930404303, −8.596331726063095723977113048279, −7.74799869060853199355491800718, −6.81029820274344770700050794901, −5.42683138880613484488478432872, −3.67791269561788982730741403459, −2.97583516384822547266443899586, −1.37943362879503725851194415219,
1.60345466189402017896303589236, 2.86100237331796493493165598900, 3.48324618989652509920690099986, 6.01752177497617732864405012531, 6.76561880358079256925236797720, 7.50881798251935506696064181864, 8.802062001746982523386452316661, 8.929399561725457578782898016351, 9.959503007938226536747586774095, 10.79469753556620195833802479452
