| L(s) = 1 | + 2·3-s − 5-s − 7-s + 9-s − 4·13-s − 2·15-s + 2·19-s − 2·21-s − 3·23-s + 25-s − 4·27-s − 9·29-s − 2·31-s + 35-s − 5·37-s − 8·39-s + 12·41-s + 5·43-s − 45-s + 12·47-s + 49-s − 9·53-s + 4·57-s − 6·59-s + 10·61-s − 63-s + 4·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.10·13-s − 0.516·15-s + 0.458·19-s − 0.436·21-s − 0.625·23-s + 1/5·25-s − 0.769·27-s − 1.67·29-s − 0.359·31-s + 0.169·35-s − 0.821·37-s − 1.28·39-s + 1.87·41-s + 0.762·43-s − 0.149·45-s + 1.75·47-s + 1/7·49-s − 1.23·53-s + 0.529·57-s − 0.781·59-s + 1.28·61-s − 0.125·63-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.728535699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.728535699\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.71874769744920, −14.21804028751006, −13.98313370185690, −13.17703636858123, −12.80972513972686, −12.18690623593887, −11.82072819915884, −10.94518206974817, −10.69278706253474, −9.681553274904188, −9.454231814622862, −9.076507190338221, −8.298762603080378, −7.814896849071149, −7.337602186702514, −7.005421434236733, −5.903246128157774, −5.592234969349406, −4.702294799916380, −3.965883799074236, −3.629380424280821, −2.784317576575199, −2.408908986768803, −1.619357384376383, −0.4238071740822535,
0.4238071740822535, 1.619357384376383, 2.408908986768803, 2.784317576575199, 3.629380424280821, 3.965883799074236, 4.702294799916380, 5.592234969349406, 5.903246128157774, 7.005421434236733, 7.337602186702514, 7.814896849071149, 8.298762603080378, 9.076507190338221, 9.454231814622862, 9.681553274904188, 10.69278706253474, 10.94518206974817, 11.82072819915884, 12.18690623593887, 12.80972513972686, 13.17703636858123, 13.98313370185690, 14.21804028751006, 14.71874769744920
