| L(s) = 1 | − 2.05·2-s − 3.41·3-s + 2.21·4-s + 5-s + 7.00·6-s − 7-s − 0.439·8-s + 8.65·9-s − 2.05·10-s − 7.55·12-s − 3.62·13-s + 2.05·14-s − 3.41·15-s − 3.52·16-s − 2.88·17-s − 17.7·18-s − 7.34·19-s + 2.21·20-s + 3.41·21-s − 2.63·23-s + 1.49·24-s + 25-s + 7.43·26-s − 19.3·27-s − 2.21·28-s + 1.64·29-s + 7.00·30-s + ⋯ |
| L(s) = 1 | − 1.45·2-s − 1.97·3-s + 1.10·4-s + 0.447·5-s + 2.86·6-s − 0.377·7-s − 0.155·8-s + 2.88·9-s − 0.649·10-s − 2.18·12-s − 1.00·13-s + 0.548·14-s − 0.881·15-s − 0.881·16-s − 0.700·17-s − 4.19·18-s − 1.68·19-s + 0.495·20-s + 0.745·21-s − 0.549·23-s + 0.305·24-s + 0.200·25-s + 1.45·26-s − 3.71·27-s − 0.418·28-s + 0.305·29-s + 1.27·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1138808461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1138808461\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.05T + 2T^{2} \) |
| 3 | \( 1 + 3.41T + 3T^{2} \) |
| 13 | \( 1 + 3.62T + 13T^{2} \) |
| 17 | \( 1 + 2.88T + 17T^{2} \) |
| 19 | \( 1 + 7.34T + 19T^{2} \) |
| 23 | \( 1 + 2.63T + 23T^{2} \) |
| 29 | \( 1 - 1.64T + 29T^{2} \) |
| 31 | \( 1 - 2.23T + 31T^{2} \) |
| 37 | \( 1 + 5.10T + 37T^{2} \) |
| 41 | \( 1 + 1.33T + 41T^{2} \) |
| 43 | \( 1 - 3.53T + 43T^{2} \) |
| 47 | \( 1 + 8.23T + 47T^{2} \) |
| 53 | \( 1 - 3.63T + 53T^{2} \) |
| 59 | \( 1 - 3.29T + 59T^{2} \) |
| 61 | \( 1 + 5.24T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 7.06T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 2.48T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 4.09T + 97T^{2} \) |
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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
−8.488765147709747570391083426421, −7.55068030613286905969457750004, −6.77249735524242498148318978949, −6.51277175301171895913795508070, −5.66328085851031692643097493376, −4.75859554413424204006475749593, −4.20253767345730163312104069058, −2.32621778350012736715354287095, −1.50386007438978487747290928993, −0.27045141369044842537986350832,
0.27045141369044842537986350832, 1.50386007438978487747290928993, 2.32621778350012736715354287095, 4.20253767345730163312104069058, 4.75859554413424204006475749593, 5.66328085851031692643097493376, 6.51277175301171895913795508070, 6.77249735524242498148318978949, 7.55068030613286905969457750004, 8.488765147709747570391083426421
