L(s) = 1 | − 3-s − 5-s + 9-s + 6·13-s + 15-s − 2·17-s − 19-s + 25-s − 27-s + 2·29-s + 4·31-s − 2·37-s − 6·39-s − 6·41-s + 4·43-s − 45-s − 7·49-s + 2·51-s − 6·53-s + 57-s − 14·61-s − 6·65-s + 4·67-s + 8·71-s − 14·73-s − 75-s − 12·79-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.66·13-s + 0.258·15-s − 0.485·17-s − 0.229·19-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.718·31-s − 0.328·37-s − 0.960·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s − 49-s + 0.280·51-s − 0.824·53-s + 0.132·57-s − 1.79·61-s − 0.744·65-s + 0.488·67-s + 0.949·71-s − 1.63·73-s − 0.115·75-s − 1.35·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−16.01177807282791, −15.58757346999694, −15.21786589441825, −14.35361014046585, −13.81774654900955, −13.26766289194021, −12.75659919793170, −12.15036536167581, −11.47844573749736, −11.19910325813923, −10.53941965012610, −10.11733864090325, −9.185603016976745, −8.719598180656412, −8.118147013914985, −7.538015926507116, −6.616230062474022, −6.368541744952892, −5.681925217746895, −4.845328040292778, −4.317404864096102, −3.592785979112246, −2.943390310655288, −1.793374976131436, −1.061817273986361, 0,
1.061817273986361, 1.793374976131436, 2.943390310655288, 3.592785979112246, 4.317404864096102, 4.845328040292778, 5.681925217746895, 6.368541744952892, 6.616230062474022, 7.538015926507116, 8.118147013914985, 8.719598180656412, 9.185603016976745, 10.11733864090325, 10.53941965012610, 11.19910325813923, 11.47844573749736, 12.15036536167581, 12.75659919793170, 13.26766289194021, 13.81774654900955, 14.35361014046585, 15.21786589441825, 15.58757346999694, 16.01177807282791