| L(s) = 1 | + 3-s − 2·4-s + 5-s − 7-s + 9-s − 3·11-s − 2·12-s − 4·13-s + 15-s + 4·16-s + 6·17-s + 19-s − 2·20-s − 21-s + 25-s + 27-s + 2·28-s + 6·29-s + 8·31-s − 3·33-s − 35-s − 2·36-s + 10·37-s − 4·39-s − 3·41-s + 10·43-s + 6·44-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.577·12-s − 1.10·13-s + 0.258·15-s + 16-s + 1.45·17-s + 0.229·19-s − 0.447·20-s − 0.218·21-s + 1/5·25-s + 0.192·27-s + 0.377·28-s + 1.11·29-s + 1.43·31-s − 0.522·33-s − 0.169·35-s − 1/3·36-s + 1.64·37-s − 0.640·39-s − 0.468·41-s + 1.52·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.510070243\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.510070243\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.18808549027958, −13.91191904954432, −13.56133938994077, −12.84138224841519, −12.50800178140907, −12.15139758977923, −11.38787043188240, −10.37760057587656, −10.21269781627352, −9.746412831641297, −9.356127223623477, −8.735000735493313, −8.034133482968831, −7.800778098098589, −7.246297027219910, −6.343281676614085, −5.849795789376670, −5.132881975746678, −4.789973331738052, −4.140799537005224, −3.349366969059214, −2.777642132236909, −2.371566289401327, −1.159816296433942, −0.6093847851134433,
0.6093847851134433, 1.159816296433942, 2.371566289401327, 2.777642132236909, 3.349366969059214, 4.140799537005224, 4.789973331738052, 5.132881975746678, 5.849795789376670, 6.343281676614085, 7.246297027219910, 7.800778098098589, 8.034133482968831, 8.735000735493313, 9.356127223623477, 9.746412831641297, 10.21269781627352, 10.37760057587656, 11.38787043188240, 12.15139758977923, 12.50800178140907, 12.84138224841519, 13.56133938994077, 13.91191904954432, 14.18808549027958
