| L(s) = 1 | + 5-s − 2·7-s − 3·11-s + 2·17-s + 19-s + 2·23-s + 25-s − 3·29-s − 3·31-s − 2·35-s + 5·41-s − 4·43-s − 8·47-s − 3·49-s + 2·53-s − 3·55-s + 3·59-s + 6·61-s − 10·67-s − 15·71-s − 14·73-s + 6·77-s − 8·79-s + 2·85-s + 89-s + 95-s − 16·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.904·11-s + 0.485·17-s + 0.229·19-s + 0.417·23-s + 1/5·25-s − 0.557·29-s − 0.538·31-s − 0.338·35-s + 0.780·41-s − 0.609·43-s − 1.16·47-s − 3/7·49-s + 0.274·53-s − 0.404·55-s + 0.390·59-s + 0.768·61-s − 1.22·67-s − 1.78·71-s − 1.63·73-s + 0.683·77-s − 0.900·79-s + 0.216·85-s + 0.105·89-s + 0.102·95-s − 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.296182208447983126213987155467, −7.46931054670746793463002543690, −6.83407958246025426756264582364, −5.88753911467151255632721223678, −5.40064086492557480332722582663, −4.43154984924062291000787946753, −3.32721269409746942774135228772, −2.69549738228437653861646620092, −1.50342828949796868296464122084, 0,
1.50342828949796868296464122084, 2.69549738228437653861646620092, 3.32721269409746942774135228772, 4.43154984924062291000787946753, 5.40064086492557480332722582663, 5.88753911467151255632721223678, 6.83407958246025426756264582364, 7.46931054670746793463002543690, 8.296182208447983126213987155467
