| L(s) = 1 | − 3.29·5-s − 2.08·7-s − 1.29·11-s + 1.21·13-s − 4.08·17-s − 19-s + 8.95·23-s + 5.87·25-s + 9.38·29-s + 6.87·35-s − 2·37-s − 3.57·41-s + 7.72·43-s − 9.46·47-s − 2.65·49-s + 11.9·53-s + 4.27·55-s + 7.21·59-s + 4.87·61-s − 3.99·65-s + 11.3·67-s + 9.02·71-s + 5.65·73-s + 2.70·77-s + 9.57·79-s − 10.7·83-s + 13.4·85-s + ⋯ |
| L(s) = 1 | − 1.47·5-s − 0.787·7-s − 0.391·11-s + 0.336·13-s − 0.990·17-s − 0.229·19-s + 1.86·23-s + 1.17·25-s + 1.74·29-s + 1.16·35-s − 0.328·37-s − 0.558·41-s + 1.17·43-s − 1.38·47-s − 0.379·49-s + 1.64·53-s + 0.576·55-s + 0.939·59-s + 0.623·61-s − 0.496·65-s + 1.39·67-s + 1.07·71-s + 0.662·73-s + 0.308·77-s + 1.07·79-s − 1.18·83-s + 1.46·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9232883795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9232883795\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 3.29T + 5T^{2} \) |
| 7 | \( 1 + 2.08T + 7T^{2} \) |
| 11 | \( 1 + 1.29T + 11T^{2} \) |
| 13 | \( 1 - 1.21T + 13T^{2} \) |
| 17 | \( 1 + 4.08T + 17T^{2} \) |
| 23 | \( 1 - 8.95T + 23T^{2} \) |
| 29 | \( 1 - 9.38T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 3.57T + 41T^{2} \) |
| 43 | \( 1 - 7.72T + 43T^{2} \) |
| 47 | \( 1 + 9.46T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 - 7.21T + 59T^{2} \) |
| 61 | \( 1 - 4.87T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 9.02T + 71T^{2} \) |
| 73 | \( 1 - 5.65T + 73T^{2} \) |
| 79 | \( 1 - 9.57T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 + 8.59T + 97T^{2} \) |
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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
−9.521343711532352671479383315856, −8.583953863782316495194189312458, −8.149447348840501420802945298796, −6.92509347355958105128872274713, −6.71421545020061515985653390695, −5.26291048552504347069671309997, −4.38188839968961964787640334971, −3.51736765501546612252135779023, −2.65747153815464978962372257134, −0.67877444607185139168919816240,
0.67877444607185139168919816240, 2.65747153815464978962372257134, 3.51736765501546612252135779023, 4.38188839968961964787640334971, 5.26291048552504347069671309997, 6.71421545020061515985653390695, 6.92509347355958105128872274713, 8.149447348840501420802945298796, 8.583953863782316495194189312458, 9.521343711532352671479383315856
