L(s) = 1 | − 4.24·5-s + 4·7-s + 16.9·11-s − 8i·13-s + 12.7i·17-s + 16i·19-s − 16.9i·23-s − 7.00·25-s − 4.24·29-s + 44·31-s − 16.9·35-s − 34i·37-s + 46.6i·41-s − 40i·43-s + 84.8i·47-s + ⋯ |
L(s) = 1 | − 0.848·5-s + 0.571·7-s + 1.54·11-s − 0.615i·13-s + 0.748i·17-s + 0.842i·19-s − 0.737i·23-s − 0.280·25-s − 0.146·29-s + 1.41·31-s − 0.484·35-s − 0.918i·37-s + 1.13i·41-s − 0.930i·43-s + 1.80i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.023189045\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.023189045\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 4.24T + 25T^{2} \) |
| 7 | \( 1 - 4T + 49T^{2} \) |
| 11 | \( 1 - 16.9T + 121T^{2} \) |
| 13 | \( 1 + 8iT - 169T^{2} \) |
| 17 | \( 1 - 12.7iT - 289T^{2} \) |
| 19 | \( 1 - 16iT - 361T^{2} \) |
| 23 | \( 1 + 16.9iT - 529T^{2} \) |
| 29 | \( 1 + 4.24T + 841T^{2} \) |
| 31 | \( 1 - 44T + 961T^{2} \) |
| 37 | \( 1 + 34iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 46.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 40iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 84.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 38.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 33.9T + 3.48e3T^{2} \) |
| 61 | \( 1 + 50iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 50.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 16T + 5.32e3T^{2} \) |
| 79 | \( 1 + 76T + 6.24e3T^{2} \) |
| 83 | \( 1 + 118.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 12.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 176T + 9.40e3T^{2} \) |
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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
−8.624065263416951622885453172415, −8.140460094738710842699741695662, −7.45357159278916529680607375986, −6.46006271500300244334404219208, −5.85428825504225885063987075588, −4.60566011930525692929212389140, −4.05133855992274238300234509729, −3.23070614190434603255454481273, −1.84468951035379798831087471219, −0.809277034049316210801965922397,
0.72569900014832694206620372434, 1.81233640217990927207400124198, 3.10570115444951735905462574386, 4.08118794736149279403699981437, 4.58037593154408979949692461879, 5.62235373311812430457537750033, 6.76706294505615217129761118980, 7.11085360055175579808827897319, 8.114021445149540307579380063089, 8.759052318427381480861237105480