| L(s) = 1 | + 5-s − 7-s − 13-s + 17-s + 3·19-s + 6·23-s + 25-s + 3·29-s − 35-s − 5·37-s + 12·41-s − 8·43-s + 12·47-s − 6·49-s − 2·53-s − 65-s − 12·67-s + 71-s − 8·73-s − 8·79-s − 83-s + 85-s − 6·89-s + 91-s + 3·95-s − 8·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 0.277·13-s + 0.242·17-s + 0.688·19-s + 1.25·23-s + 1/5·25-s + 0.557·29-s − 0.169·35-s − 0.821·37-s + 1.87·41-s − 1.21·43-s + 1.75·47-s − 6/7·49-s − 0.274·53-s − 0.124·65-s − 1.46·67-s + 0.118·71-s − 0.936·73-s − 0.900·79-s − 0.109·83-s + 0.108·85-s − 0.635·89-s + 0.104·91-s + 0.307·95-s − 0.812·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.18955619344981, −13.53222097356002, −13.28467736121813, −12.65657446668384, −12.20232843815889, −11.80996580127994, −10.97563289990816, −10.82371942439987, −9.990352510255389, −9.793513844398380, −9.094866941432975, −8.799488064810218, −8.113429981392033, −7.396992130519727, −7.122250581323864, −6.487874375092668, −5.874067006746039, −5.441939039997884, −4.823275220492842, −4.292898183893579, −3.493821132216352, −2.915861908148212, −2.506790041075343, −1.522066074368016, −1.003774005379177, 0,
1.003774005379177, 1.522066074368016, 2.506790041075343, 2.915861908148212, 3.493821132216352, 4.292898183893579, 4.823275220492842, 5.441939039997884, 5.874067006746039, 6.487874375092668, 7.122250581323864, 7.396992130519727, 8.113429981392033, 8.799488064810218, 9.094866941432975, 9.793513844398380, 9.990352510255389, 10.82371942439987, 10.97563289990816, 11.80996580127994, 12.20232843815889, 12.65657446668384, 13.28467736121813, 13.53222097356002, 14.18955619344981
