| L(s) = 1 | + 7-s − 6·11-s + 5·13-s − 3·17-s − 19-s + 3·23-s − 5·25-s − 9·29-s + 4·31-s + 2·37-s − 8·43-s − 6·49-s + 3·53-s + 9·59-s − 10·61-s − 5·67-s − 6·71-s − 7·73-s − 6·77-s + 10·79-s − 6·83-s + 12·89-s + 5·91-s − 10·97-s − 18·101-s − 14·103-s − 9·107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.80·11-s + 1.38·13-s − 0.727·17-s − 0.229·19-s + 0.625·23-s − 25-s − 1.67·29-s + 0.718·31-s + 0.328·37-s − 1.21·43-s − 6/7·49-s + 0.412·53-s + 1.17·59-s − 1.28·61-s − 0.610·67-s − 0.712·71-s − 0.819·73-s − 0.683·77-s + 1.12·79-s − 0.658·83-s + 1.27·89-s + 0.524·91-s − 1.01·97-s − 1.79·101-s − 1.37·103-s − 0.870·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−8.297610084688077297742296971210, −7.87841813427368117647214734878, −6.98043076884761860510728876245, −6.02372043714400021098795022997, −5.38865185009354292107305454716, −4.54275466905147962254344391783, −3.59163897364434449988676374845, −2.60234956115962506172119076503, −1.59940757777111675726084768164, 0,
1.59940757777111675726084768164, 2.60234956115962506172119076503, 3.59163897364434449988676374845, 4.54275466905147962254344391783, 5.38865185009354292107305454716, 6.02372043714400021098795022997, 6.98043076884761860510728876245, 7.87841813427368117647214734878, 8.297610084688077297742296971210
