| L(s) = 1 | + 0.331·5-s + 3.08·7-s − 3.69·11-s + 4.64·13-s − 6.52·17-s + 0.867·19-s − 4·23-s − 4.88·25-s + 4.89·29-s + 6.14·31-s + 1.02·35-s − 3.64·37-s − 3.92·41-s − 3.92·43-s + 1.65·47-s + 2.50·49-s − 0.564·53-s − 1.22·55-s − 6.59·59-s − 14.8·61-s + 1.53·65-s − 13.9·67-s − 7.49·71-s + 5.62·73-s − 11.3·77-s + 14.9·79-s + 9.35·83-s + ⋯ |
| L(s) = 1 | + 0.148·5-s + 1.16·7-s − 1.11·11-s + 1.28·13-s − 1.58·17-s + 0.198·19-s − 0.834·23-s − 0.977·25-s + 0.908·29-s + 1.10·31-s + 0.172·35-s − 0.599·37-s − 0.613·41-s − 0.599·43-s + 0.241·47-s + 0.357·49-s − 0.0775·53-s − 0.165·55-s − 0.858·59-s − 1.89·61-s + 0.190·65-s − 1.70·67-s − 0.889·71-s + 0.658·73-s − 1.29·77-s + 1.67·79-s + 1.02·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 0.331T + 5T^{2} \) |
| 7 | \( 1 - 3.08T + 7T^{2} \) |
| 11 | \( 1 + 3.69T + 11T^{2} \) |
| 13 | \( 1 - 4.64T + 13T^{2} \) |
| 17 | \( 1 + 6.52T + 17T^{2} \) |
| 19 | \( 1 - 0.867T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 4.89T + 29T^{2} \) |
| 31 | \( 1 - 6.14T + 31T^{2} \) |
| 37 | \( 1 + 3.64T + 37T^{2} \) |
| 41 | \( 1 + 3.92T + 41T^{2} \) |
| 43 | \( 1 + 3.92T + 43T^{2} \) |
| 47 | \( 1 - 1.65T + 47T^{2} \) |
| 53 | \( 1 + 0.564T + 53T^{2} \) |
| 59 | \( 1 + 6.59T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 7.49T + 71T^{2} \) |
| 73 | \( 1 - 5.62T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 9.35T + 83T^{2} \) |
| 89 | \( 1 - 18.1T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−7.63024928808170003597930362576, −6.49613441365902623193251383087, −6.14187175139525447578243528478, −5.17929021222829505231545324668, −4.66872830113867590151973803393, −3.96867063721958662060878579118, −2.95494517854485938188432057147, −2.08783694473472480641157552017, −1.39770654515595401705486031378, 0,
1.39770654515595401705486031378, 2.08783694473472480641157552017, 2.95494517854485938188432057147, 3.96867063721958662060878579118, 4.66872830113867590151973803393, 5.17929021222829505231545324668, 6.14187175139525447578243528478, 6.49613441365902623193251383087, 7.63024928808170003597930362576
