| L(s) = 1 | + 3-s − 3.33i·5-s + (−1.56 + 2.13i)7-s + 9-s − 0.936i·11-s − 1.87i·13-s − 3.33i·15-s − 5.20i·17-s − 7.12·19-s + (−1.56 + 2.13i)21-s − 0.936i·23-s − 6.12·25-s + 27-s + 2·29-s − 0.936i·33-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.49i·5-s + (−0.590 + 0.807i)7-s + 0.333·9-s − 0.282i·11-s − 0.519i·13-s − 0.861i·15-s − 1.26i·17-s − 1.63·19-s + (−0.340 + 0.466i)21-s − 0.195i·23-s − 1.22·25-s + 0.192·27-s + 0.371·29-s − 0.163i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.335224780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.335224780\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + (1.56 - 2.13i)T \) |
| good | 5 | \( 1 + 3.33iT - 5T^{2} \) |
| 11 | \( 1 + 0.936iT - 11T^{2} \) |
| 13 | \( 1 + 1.87iT - 13T^{2} \) |
| 17 | \( 1 + 5.20iT - 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 + 0.936iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 1.12T + 37T^{2} \) |
| 41 | \( 1 + 1.46iT - 41T^{2} \) |
| 43 | \( 1 + 9.06iT - 43T^{2} \) |
| 47 | \( 1 - 6.24T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 4.79iT - 61T^{2} \) |
| 67 | \( 1 - 10.9iT - 67T^{2} \) |
| 71 | \( 1 + 3.86iT - 71T^{2} \) |
| 73 | \( 1 - 6.67iT - 73T^{2} \) |
| 79 | \( 1 + 2.39iT - 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 1.46iT - 89T^{2} \) |
| 97 | \( 1 + 10.4iT - 97T^{2} \) |
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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
−9.098513752921468604131295852044, −8.677431865742849238179189542218, −8.033211577016188628109206975066, −6.89036646959552419444126580089, −5.87561353580436417830888764364, −5.07410898654795042365793588850, −4.24333295336788032163965425361, −3.05273972029844201553494642907, −2.02561978360201775546841953808, −0.48262282007089484425203160717,
1.83274206804510419991550996831, 2.88910444178870556983193640805, 3.74491467132211435780822346536, 4.46726537021177509265900310762, 6.35491217561144620350206632060, 6.44388256344380015865857769981, 7.44295376892471482179076549177, 8.129337866645383739150812192074, 9.195253782614435932270828601096, 10.00369747188938389394246393713
