| L(s) = 1 | − 2·3-s − 2·7-s + 9-s − 4·17-s + 4·19-s + 4·21-s − 6·23-s + 4·27-s − 2·29-s − 8·31-s + 4·37-s + 6·41-s + 6·43-s − 2·47-s − 3·49-s + 8·51-s + 12·53-s − 8·57-s − 4·59-s − 14·61-s − 2·63-s + 10·67-s + 12·69-s − 8·71-s + 4·73-s − 8·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s + 1/3·9-s − 0.970·17-s + 0.917·19-s + 0.872·21-s − 1.25·23-s + 0.769·27-s − 0.371·29-s − 1.43·31-s + 0.657·37-s + 0.937·41-s + 0.914·43-s − 0.291·47-s − 3/7·49-s + 1.12·51-s + 1.64·53-s − 1.05·57-s − 0.520·59-s − 1.79·61-s − 0.251·63-s + 1.22·67-s + 1.44·69-s − 0.949·71-s + 0.468·73-s − 0.900·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−14.84713006570817, −14.19752099296196, −13.82933594496185, −13.04515977770796, −12.79181124734881, −12.16609055437646, −11.75449415612089, −11.05913271962897, −10.97034395119631, −10.14076674369307, −9.731466034305177, −9.123120317131945, −8.667779326710926, −7.768989821202820, −7.324810634586470, −6.735165521217686, −6.026880143497658, −5.871233803281569, −5.222526560854363, −4.474436745469438, −3.969448156952587, −3.201228770911189, −2.489588721801794, −1.677936242987862, −0.6817185395730227, 0,
0.6817185395730227, 1.677936242987862, 2.489588721801794, 3.201228770911189, 3.969448156952587, 4.474436745469438, 5.222526560854363, 5.871233803281569, 6.026880143497658, 6.735165521217686, 7.324810634586470, 7.768989821202820, 8.667779326710926, 9.123120317131945, 9.731466034305177, 10.14076674369307, 10.97034395119631, 11.05913271962897, 11.75449415612089, 12.16609055437646, 12.79181124734881, 13.04515977770796, 13.82933594496185, 14.19752099296196, 14.84713006570817
