| L(s) = 1 | + 2·3-s − 5-s − 2·7-s + 9-s − 2·13-s − 2·15-s + 6·17-s + 4·19-s − 4·21-s + 6·23-s + 25-s − 4·27-s − 6·29-s − 4·31-s + 2·35-s + 2·37-s − 4·39-s − 10·43-s − 45-s + 6·47-s − 3·49-s + 12·51-s + 6·53-s + 8·57-s + 12·59-s + 2·61-s − 2·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.554·13-s − 0.516·15-s + 1.45·17-s + 0.917·19-s − 0.872·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s − 0.718·31-s + 0.338·35-s + 0.328·37-s − 0.640·39-s − 1.52·43-s − 0.149·45-s + 0.875·47-s − 3/7·49-s + 1.68·51-s + 0.824·53-s + 1.05·57-s + 1.56·59-s + 0.256·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.646559745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.646559745\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 41 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.83926008853248, −14.62820011161943, −14.02106614066436, −13.40926326654343, −12.97629092108170, −12.52281783806882, −11.77651950368918, −11.45858224962783, −10.65257383470915, −9.946945471924193, −9.589760023860822, −9.129085348602566, −8.523741392427034, −7.935882539499942, −7.368360122672907, −7.110310273432995, −6.236190306506292, −5.317701297494548, −5.163787655080010, −3.875236943029816, −3.603101880327984, −3.012232794678947, −2.479677360106350, −1.519168719873639, −0.5735767811929239,
0.5735767811929239, 1.519168719873639, 2.479677360106350, 3.012232794678947, 3.603101880327984, 3.875236943029816, 5.163787655080010, 5.317701297494548, 6.236190306506292, 7.110310273432995, 7.368360122672907, 7.935882539499942, 8.523741392427034, 9.129085348602566, 9.589760023860822, 9.946945471924193, 10.65257383470915, 11.45858224962783, 11.77651950368918, 12.52281783806882, 12.97629092108170, 13.40926326654343, 14.02106614066436, 14.62820011161943, 14.83926008853248
