| L(s) = 1 | + (−0.415 + 0.909i)3-s + (0.443 + 0.130i)5-s + (4.44 − 0.856i)7-s + (−0.654 − 0.755i)9-s + (0.933 − 0.890i)11-s + (−0.196 + 4.13i)13-s + (−0.302 + 0.349i)15-s + (4.79 − 3.77i)17-s + (−7.26 − 1.40i)19-s + (−1.06 + 4.39i)21-s + (6.65 − 0.635i)23-s + (−4.02 − 2.58i)25-s + (0.959 − 0.281i)27-s + (3.87 − 6.71i)29-s + (0.116 + 2.44i)31-s + ⋯ |
| L(s) = 1 | + (−0.239 + 0.525i)3-s + (0.198 + 0.0582i)5-s + (1.67 − 0.323i)7-s + (−0.218 − 0.251i)9-s + (0.281 − 0.268i)11-s + (−0.0545 + 1.14i)13-s + (−0.0782 + 0.0902i)15-s + (1.16 − 0.915i)17-s + (−1.66 − 0.321i)19-s + (−0.232 + 0.959i)21-s + (1.38 − 0.132i)23-s + (−0.805 − 0.517i)25-s + (0.184 − 0.0542i)27-s + (0.720 − 1.24i)29-s + (0.0208 + 0.438i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.77285 + 0.324854i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.77285 + 0.324854i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.415 - 0.909i)T \) |
| 67 | \( 1 + (-4.28 - 6.97i)T \) |
| good | 5 | \( 1 + (-0.443 - 0.130i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (-4.44 + 0.856i)T + (6.49 - 2.60i)T^{2} \) |
| 11 | \( 1 + (-0.933 + 0.890i)T + (0.523 - 10.9i)T^{2} \) |
| 13 | \( 1 + (0.196 - 4.13i)T + (-12.9 - 1.23i)T^{2} \) |
| 17 | \( 1 + (-4.79 + 3.77i)T + (4.00 - 16.5i)T^{2} \) |
| 19 | \( 1 + (7.26 + 1.40i)T + (17.6 + 7.06i)T^{2} \) |
| 23 | \( 1 + (-6.65 + 0.635i)T + (22.5 - 4.35i)T^{2} \) |
| 29 | \( 1 + (-3.87 + 6.71i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.116 - 2.44i)T + (-30.8 + 2.94i)T^{2} \) |
| 37 | \( 1 + (-4.26 - 7.38i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.94 + 0.779i)T + (29.6 + 28.2i)T^{2} \) |
| 43 | \( 1 + (-0.606 - 4.22i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (-6.09 - 8.56i)T + (-15.3 + 44.4i)T^{2} \) |
| 53 | \( 1 + (1.36 - 9.51i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (3.19 - 2.05i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (5.68 + 5.42i)T + (2.90 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-8.96 - 7.05i)T + (16.7 + 68.9i)T^{2} \) |
| 73 | \( 1 + (-1.07 - 1.02i)T + (3.47 + 72.9i)T^{2} \) |
| 79 | \( 1 + (-0.150 + 0.0777i)T + (45.8 - 64.3i)T^{2} \) |
| 83 | \( 1 + (1.28 + 5.30i)T + (-73.7 + 38.0i)T^{2} \) |
| 89 | \( 1 + (7.14 + 15.6i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (8.15 + 14.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.39193817682686689884377511030, −9.492582204131638293957426863862, −8.621107891680072700712908872419, −7.86362058107600144859936899472, −6.81377179712329872169920818635, −5.82887234875145665908078894575, −4.61454988323717372243493869646, −4.36842305220037805447328100436, −2.66680236085183662409573776632, −1.26438323686313856582029339065,
1.26063732067805431138899037274, 2.24664414647146848576610681075, 3.82301807268968326296330756865, 5.12670437629166532694181990275, 5.60177089074412919257163251379, 6.76979837794288949189074446145, 7.898165483839250735710277211893, 8.216662483922056023176464607797, 9.224416499063772527253831872700, 10.66504425659023718248082930477
