| L(s) = 1 | + 0.520·3-s + 2.23i·5-s + 10.3i·7-s − 8.72·9-s − 0.821i·11-s − 9.69·13-s + 1.16i·15-s + 21.0i·17-s − 4.31i·19-s + 5.37i·21-s + (−20.2 − 10.9i)23-s − 5.00·25-s − 9.22·27-s + 19.5·29-s − 7.54·31-s + ⋯ |
| L(s) = 1 | + 0.173·3-s + 0.447i·5-s + 1.47i·7-s − 0.969·9-s − 0.0746i·11-s − 0.745·13-s + 0.0776i·15-s + 1.23i·17-s − 0.227i·19-s + 0.255i·21-s + (−0.879 − 0.475i)23-s − 0.200·25-s − 0.341·27-s + 0.675·29-s − 0.243·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.475 + 0.879i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.001241769719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.001241769719\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (20.2 + 10.9i)T \) |
| good | 3 | \( 1 - 0.520T + 9T^{2} \) |
| 7 | \( 1 - 10.3iT - 49T^{2} \) |
| 11 | \( 1 + 0.821iT - 121T^{2} \) |
| 13 | \( 1 + 9.69T + 169T^{2} \) |
| 17 | \( 1 - 21.0iT - 289T^{2} \) |
| 19 | \( 1 + 4.31iT - 361T^{2} \) |
| 29 | \( 1 - 19.5T + 841T^{2} \) |
| 31 | \( 1 + 7.54T + 961T^{2} \) |
| 37 | \( 1 + 3.21iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 40.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + 62.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 57.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + 20.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 51.3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 61.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 50.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 38.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 0.482T + 5.32e3T^{2} \) |
| 79 | \( 1 + 1.75iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 112. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 42.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 47.0iT - 9.40e3T^{2} \) |
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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
−8.604468902106061090192614991589, −8.335625406289142165926717011378, −7.26040506823483305694948408115, −6.14932906408127753897631090101, −5.80054516929915769240014148014, −4.81503069240676276940659445766, −3.59048049898028387816759986008, −2.63014735168375400129716740752, −2.03346085955297139428098305159, −0.00033210489355954367688232178,
1.06033316067967595094540912413, 2.47433076785585543738818587020, 3.47518249370832430209146572304, 4.43225634042570793421321150598, 5.13899564155428463840833738508, 6.16165695713115287930646321879, 7.12079050375977356786175767586, 7.75900362088245812522018002998, 8.410640178315445731492731385096, 9.545101068727681072492149283755
