| L(s) = 1 | − 8.31i·3-s − 1.48i·5-s − 13.7i·7-s + 11.7·9-s + 10.2·11-s + 98.4·13-s − 12.3·15-s − 286.·17-s − 367. i·19-s − 114.·21-s + 242.·23-s + 622.·25-s − 771. i·27-s + 1.14e3i·29-s − 895.·31-s + ⋯ |
| L(s) = 1 | − 0.924i·3-s − 0.0592i·5-s − 0.281i·7-s + 0.145·9-s + 0.0843·11-s + 0.582·13-s − 0.0548·15-s − 0.991·17-s − 1.01i·19-s − 0.260·21-s + 0.457·23-s + 0.996·25-s − 1.05i·27-s + 1.36i·29-s − 0.931·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.759292550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.759292550\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + (-1.54e3 + 1.01e3i)T \) |
| good | 3 | \( 1 + 8.31iT - 81T^{2} \) |
| 5 | \( 1 + 1.48iT - 625T^{2} \) |
| 7 | \( 1 + 13.7iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 10.2T + 1.46e4T^{2} \) |
| 13 | \( 1 - 98.4T + 2.85e4T^{2} \) |
| 17 | \( 1 + 286.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 367. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 242.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 1.14e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 895.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 2.29e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.69e3T + 2.82e6T^{2} \) |
| 47 | \( 1 + 743.T + 4.87e6T^{2} \) |
| 53 | \( 1 - 99.3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 3.28e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 3.22e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 5.55e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 2.95e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 3.61e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 2.38e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 6.42e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 3.29e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 9.32e3T + 8.85e7T^{2} \) |
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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
−9.226417477715849846392084031580, −8.794136743039458618180646292124, −7.50001926888764595519296637887, −7.03076062321494492077257159263, −6.18494215173837884868022721739, −4.97565048923270410278710849449, −3.92956204057052920766793872915, −2.59236939581428303492805728274, −1.46549029822792249269872381594, −0.44113368928918124714758944241,
1.30300736645609123049108722513, 2.73392232684809824982695743768, 3.90679415568179130175381240229, 4.59514513503226818787881208531, 5.69751352314797031137362999567, 6.60488414312161941382459932546, 7.70839986971803605440475625716, 8.754875764118318030971842946261, 9.367551747287176548747707103229, 10.27770570145712116224860046415
