| L(s) = 1 | + 3.31·3-s − 3i·5-s + 8·9-s + 3.31·11-s − 9.94i·15-s + 3.31i·23-s − 4·25-s + 16.5·27-s + 9.94i·31-s + 11·33-s − 7i·37-s − 24i·45-s − 6.63i·47-s − 7·49-s − 6i·53-s + ⋯ |
| L(s) = 1 | + 1.91·3-s − 1.34i·5-s + 2.66·9-s + 1.00·11-s − 2.56i·15-s + 0.691i·23-s − 0.800·25-s + 3.19·27-s + 1.78i·31-s + 1.91·33-s − 1.15i·37-s − 3.57i·45-s − 0.967i·47-s − 49-s − 0.824i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.164253662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.164253662\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - 3.31T \) |
| good | 3 | \( 1 - 3.31T + 3T^{2} \) |
| 5 | \( 1 + 3iT - 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 3.31iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 9.94iT - 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 6.63iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 3.31T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 9.94T + 67T^{2} \) |
| 71 | \( 1 - 16.5iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 + 17T + 97T^{2} \) |
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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
−8.594219145927245831589693873961, −8.346137031523468576610632620350, −7.37106032732570703157523247940, −6.74073489459835952980295578983, −5.40332961934681688588666639330, −4.51761137104174719945196121106, −3.84820595234986949428889067284, −3.09277439014469618477807125665, −1.86137532145363574658900962186, −1.21986679994038940261050732527,
1.52562431655728173306011074383, 2.52183389423270904862779536715, 3.08465445247433503966741023318, 3.86754329358391859924914604225, 4.56400322265379211794966593772, 6.28112950243654580729310155764, 6.69706700067729986285344019398, 7.63748514795801563898215215484, 7.973261088010321470903721658248, 9.004101929207985475408220972290
