| L(s) = 1 | − 4·11-s + 3·13-s + 7·17-s − 6·19-s + 9·23-s + 3·29-s − 7·31-s − 10·37-s + 41-s − 13·43-s + 2·47-s + 53-s − 11·59-s − 13·61-s − 8·71-s − 8·73-s − 4·79-s − 7·83-s + 14·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 0.832·13-s + 1.69·17-s − 1.37·19-s + 1.87·23-s + 0.557·29-s − 1.25·31-s − 1.64·37-s + 0.156·41-s − 1.98·43-s + 0.291·47-s + 0.137·53-s − 1.43·59-s − 1.66·61-s − 0.949·71-s − 0.936·73-s − 0.450·79-s − 0.768·83-s + 1.48·89-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.152278788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.152278788\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−13.22961684385942, −12.67752609340712, −12.35284070670515, −11.85180712781096, −11.13810256476000, −10.74154166385054, −10.43057828044549, −10.05774900518101, −9.256417174995336, −8.846466782954940, −8.431791983251850, −7.900235974739446, −7.421434537286936, −6.925839552876291, −6.362736183454912, −5.767595027100107, −5.287499575833935, −4.917862204430374, −4.262640256823690, −3.464254689302393, −3.156372012353280, −2.637869646632564, −1.625059779750392, −1.383781305891869, −0.3041149958329627,
0.3041149958329627, 1.383781305891869, 1.625059779750392, 2.637869646632564, 3.156372012353280, 3.464254689302393, 4.262640256823690, 4.917862204430374, 5.287499575833935, 5.767595027100107, 6.362736183454912, 6.925839552876291, 7.421434537286936, 7.900235974739446, 8.431791983251850, 8.846466782954940, 9.256417174995336, 10.05774900518101, 10.43057828044549, 10.74154166385054, 11.13810256476000, 11.85180712781096, 12.35284070670515, 12.67752609340712, 13.22961684385942
