| L(s) = 1 | − 3-s − 2·5-s + 7-s + 9-s − 6·13-s + 2·15-s − 17-s − 2·19-s − 21-s − 25-s − 27-s − 8·29-s − 2·35-s − 2·37-s + 6·39-s + 2·41-s + 8·43-s − 2·45-s + 8·47-s + 49-s + 51-s + 2·53-s + 2·57-s + 12·59-s + 4·61-s + 63-s + 12·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.66·13-s + 0.516·15-s − 0.242·17-s − 0.458·19-s − 0.218·21-s − 1/5·25-s − 0.192·27-s − 1.48·29-s − 0.338·35-s − 0.328·37-s + 0.960·39-s + 0.312·41-s + 1.21·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s + 0.140·51-s + 0.274·53-s + 0.264·57-s + 1.56·59-s + 0.512·61-s + 0.125·63-s + 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−15.72057236116047, −15.24926287841839, −14.72434340561615, −14.34263629628976, −13.55127591118594, −12.84576381121543, −12.42966012201536, −11.92526971202685, −11.45441173948055, −10.93086552464108, −10.40612193654477, −9.677512172017061, −9.217772274889937, −8.440557677876152, −7.739927519320670, −7.400628014243907, −6.874359198677980, −6.064868444241493, −5.277255648013163, −4.954198550766992, −3.990286467246042, −3.839504897391624, −2.515994038434976, −2.077022901432816, −0.8009982056744012, 0,
0.8009982056744012, 2.077022901432816, 2.515994038434976, 3.839504897391624, 3.990286467246042, 4.954198550766992, 5.277255648013163, 6.064868444241493, 6.874359198677980, 7.400628014243907, 7.739927519320670, 8.440557677876152, 9.217772274889937, 9.677512172017061, 10.40612193654477, 10.93086552464108, 11.45441173948055, 11.92526971202685, 12.42966012201536, 12.84576381121543, 13.55127591118594, 14.34263629628976, 14.72434340561615, 15.24926287841839, 15.72057236116047
