| L(s) = 1 | + (−0.173 − 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)5-s + (−0.766 − 0.642i)6-s + (0.326 − 0.565i)7-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)9-s + (0.173 − 0.984i)10-s + (−1.78 − 3.09i)11-s + (−0.499 + 0.866i)12-s + (−3.61 − 3.03i)13-s + (−0.613 − 0.223i)14-s + (0.939 − 0.342i)15-s + (0.766 − 0.642i)16-s + (0.0393 + 0.223i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (0.442 − 0.371i)3-s + (−0.469 + 0.171i)4-s + (0.420 + 0.152i)5-s + (−0.312 − 0.262i)6-s + (0.123 − 0.213i)7-s + (0.176 + 0.306i)8-s + (0.0578 − 0.328i)9-s + (0.0549 − 0.311i)10-s + (−0.538 − 0.933i)11-s + (−0.144 + 0.249i)12-s + (−1.00 − 0.840i)13-s + (−0.163 − 0.0596i)14-s + (0.242 − 0.0883i)15-s + (0.191 − 0.160i)16-s + (0.00954 + 0.0541i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 + 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.473 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.761515 - 1.27400i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.761515 - 1.27400i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 3 | \( 1 + (-0.766 + 0.642i)T \) |
| 5 | \( 1 + (-0.939 - 0.342i)T \) |
| 19 | \( 1 + (-4.07 + 1.55i)T \) |
| good | 7 | \( 1 + (-0.326 + 0.565i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.78 + 3.09i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.61 + 3.03i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.0393 - 0.223i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-8.55 + 3.11i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.58 + 8.97i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (2.40 - 4.17i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 1.36T + 37T^{2} \) |
| 41 | \( 1 + (-5.06 + 4.25i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (11.5 + 4.20i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (2.04 - 11.6i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (5.29 - 1.92i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.54 - 8.74i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-0.333 + 0.121i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (2.09 - 11.9i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-11.4 - 4.15i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-2.31 + 1.94i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (7.71 - 6.47i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-8.69 + 15.0i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.44 - 4.56i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.0949 - 0.538i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.48484461549658878970035194407, −9.645007534504098942054064277654, −8.787319915676004367711424893843, −7.88549190677733119617301814493, −7.06475202448203268439256590387, −5.71724256850871206206075036705, −4.76698957279706603935881082885, −3.19230474073867697764224246501, −2.56367965758591254670156197830, −0.873891891203579807056629521243,
1.87816244016718787915792965853, 3.32507181093303288893358341960, 4.96676712884316885146914532081, 5.10944609413149526633166910534, 6.75908751802252320710916440794, 7.38674260542462483607850816355, 8.390002484042593734292020051153, 9.499257504964784300406269630224, 9.629501854940229728244449691559, 10.81285942109018589532843016964
