L(s) = 1 | + (−0.277 + 1.70i)3-s + (0.459 − 0.459i)5-s − 7-s + (−2.84 − 0.947i)9-s + (−1.43 − 1.43i)11-s + (−1.18 + 1.18i)13-s + (0.658 + 0.912i)15-s − 7.20i·17-s + (2.23 + 2.23i)19-s + (0.277 − 1.70i)21-s − 0.540i·23-s + 4.57i·25-s + (2.40 − 4.60i)27-s + (−4.14 − 4.14i)29-s − 10.4i·31-s + ⋯ |
L(s) = 1 | + (−0.159 + 0.987i)3-s + (0.205 − 0.205i)5-s − 0.377·7-s + (−0.948 − 0.315i)9-s + (−0.432 − 0.432i)11-s + (−0.328 + 0.328i)13-s + (0.169 + 0.235i)15-s − 1.74i·17-s + (0.512 + 0.512i)19-s + (0.0604 − 0.373i)21-s − 0.112i·23-s + 0.915i·25-s + (0.463 − 0.886i)27-s + (−0.769 − 0.769i)29-s − 1.87i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8562808877\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8562808877\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.277 - 1.70i)T \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + (-0.459 + 0.459i)T - 5iT^{2} \) |
| 11 | \( 1 + (1.43 + 1.43i)T + 11iT^{2} \) |
| 13 | \( 1 + (1.18 - 1.18i)T - 13iT^{2} \) |
| 17 | \( 1 + 7.20iT - 17T^{2} \) |
| 19 | \( 1 + (-2.23 - 2.23i)T + 19iT^{2} \) |
| 23 | \( 1 + 0.540iT - 23T^{2} \) |
| 29 | \( 1 + (4.14 + 4.14i)T + 29iT^{2} \) |
| 31 | \( 1 + 10.4iT - 31T^{2} \) |
| 37 | \( 1 + (1.36 + 1.36i)T + 37iT^{2} \) |
| 41 | \( 1 + 2.16T + 41T^{2} \) |
| 43 | \( 1 + (-6.40 + 6.40i)T - 43iT^{2} \) |
| 47 | \( 1 + 7.63T + 47T^{2} \) |
| 53 | \( 1 + (-2.29 + 2.29i)T - 53iT^{2} \) |
| 59 | \( 1 + (-6.32 - 6.32i)T + 59iT^{2} \) |
| 61 | \( 1 + (-0.637 + 0.637i)T - 61iT^{2} \) |
| 67 | \( 1 + (-2.14 - 2.14i)T + 67iT^{2} \) |
| 71 | \( 1 + 14.6iT - 71T^{2} \) |
| 73 | \( 1 + 0.233iT - 73T^{2} \) |
| 79 | \( 1 - 4.93iT - 79T^{2} \) |
| 83 | \( 1 + (1.76 - 1.76i)T - 83iT^{2} \) |
| 89 | \( 1 + 7.79T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−9.600064329399478749881618001354, −8.945921361197171430869727655635, −7.87825554451181177313781396602, −7.03014624545493009211956649945, −5.79622804160907717733964685178, −5.34643887390025088150961835362, −4.33436375084532774464389709163, −3.38859074064925225643138671489, −2.40088894795459474681644698861, −0.35998077027566232532996319656,
1.38894075149297121902003459853, 2.49415165483447635360449048175, 3.47436184367037745850404906325, 4.90489800041949210813005917101, 5.75573138360582730102079704006, 6.58583643349903240726755665208, 7.21142975372725341791062421566, 8.116264521728420736199945358119, 8.751560485844781961490610884956, 9.883027789505703847299710446715