| L(s) = 1 | + i·5-s + 2.44i·7-s + 5.91·11-s + 6.24·13-s − 4.89i·19-s + 1.43·23-s − 25-s + 6i·29-s + 1.43i·31-s − 2.44·35-s + 2.24·37-s − 4.24i·41-s − 11.8i·43-s − 11.8·47-s + 1.00·49-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s + 0.925i·7-s + 1.78·11-s + 1.73·13-s − 1.12i·19-s + 0.299·23-s − 0.200·25-s + 1.11i·29-s + 0.257i·31-s − 0.414·35-s + 0.368·37-s − 0.662i·41-s − 1.80i·43-s − 1.72·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.326989694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.326989694\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| good | 7 | \( 1 - 2.44iT - 7T^{2} \) |
| 11 | \( 1 - 5.91T + 11T^{2} \) |
| 13 | \( 1 - 6.24T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 4.89iT - 19T^{2} \) |
| 23 | \( 1 - 1.43T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 1.43iT - 31T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 + 4.24iT - 41T^{2} \) |
| 43 | \( 1 + 11.8iT - 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 + 8.48iT - 53T^{2} \) |
| 59 | \( 1 - 5.91T + 59T^{2} \) |
| 61 | \( 1 - 6.48T + 61T^{2} \) |
| 67 | \( 1 - 6.92iT - 67T^{2} \) |
| 71 | \( 1 + 2.86T + 71T^{2} \) |
| 73 | \( 1 - 6.48T + 73T^{2} \) |
| 79 | \( 1 + 8.36iT - 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 7.75iT - 89T^{2} \) |
| 97 | \( 1 - 6.48T + 97T^{2} \) |
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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
−8.806747363629074668140114372316, −8.436850662215661561315764387003, −7.02767270732029429574263524497, −6.65026879622332761483176281365, −5.89880981357703271262227736024, −5.06285498148384800744256473338, −3.87389847247145485961345144331, −3.36122959050099884404823524643, −2.14978452676469182857754390828, −1.10480550911103340119463080808,
1.00776588218182398215731759890, 1.56968382892887933190662398362, 3.30942444416256878990616694646, 4.01208680291662378199806808265, 4.48827740997003939296359939461, 5.93972311887789730513803773817, 6.26815764478723256034007417959, 7.14211283226350814429624228907, 8.141551349996605813402789936593, 8.550628123382656923895633277093
