| L(s) = 1 | + (−1 − 1.41i)3-s + 1.41i·5-s − 1.41i·7-s + (−1.00 + 2.82i)9-s + 4·11-s + 13-s + (2.00 − 1.41i)15-s − 4.24i·19-s + (−2.00 + 1.41i)21-s + 2·23-s + 2.99·25-s + (5.00 − 1.41i)27-s − 8.48i·29-s − 1.41i·31-s + (−4 − 5.65i)33-s + ⋯ |
| L(s) = 1 | + (−0.577 − 0.816i)3-s + 0.632i·5-s − 0.534i·7-s + (−0.333 + 0.942i)9-s + 1.20·11-s + 0.277·13-s + (0.516 − 0.365i)15-s − 0.973i·19-s + (−0.436 + 0.308i)21-s + 0.417·23-s + 0.599·25-s + (0.962 − 0.272i)27-s − 1.57i·29-s − 0.254i·31-s + (−0.696 − 0.984i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12721 - 0.583489i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12721 - 0.583489i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 + 1.41i)T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 1.41iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 4.24iT - 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + 8.48iT - 29T^{2} \) |
| 31 | \( 1 + 1.41iT - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 7.07iT - 41T^{2} \) |
| 43 | \( 1 + 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 5.65iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 4.24iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 2.82iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 9.89iT - 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−10.70918683187276525811384195475, −9.710350488402239538089857916654, −8.630231414202416579819714795073, −7.60305565572222504740081702585, −6.74544073798919431085558828688, −6.32893401079110007355409021257, −5.02072506366081317537826705701, −3.82613565505488535125785806810, −2.42154474699420903652764897984, −0.918931812453544451977280289342,
1.29172349976501918609357431475, 3.27812021251300905088686748338, 4.28004962764815020424843028154, 5.22201007029872046677461788466, 6.04702929474336970967792349551, 6.98913781088046942509248026430, 8.542846653314757228562986559027, 8.991459345817929650390101653036, 9.822996393341571681494238268963, 10.76680887051288588674019510394
