| L(s) = 1 | + 3-s − 5-s + 9-s − 2·11-s − 4·13-s − 15-s − 2·17-s − 2·19-s + 4·23-s + 25-s + 27-s + 6·29-s + 2·31-s − 2·33-s + 10·37-s − 4·39-s + 10·41-s + 12·43-s − 45-s + 8·47-s − 2·51-s + 2·55-s − 2·57-s + 8·59-s + 2·61-s + 4·65-s − 12·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 0.258·15-s − 0.485·17-s − 0.458·19-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.359·31-s − 0.348·33-s + 1.64·37-s − 0.640·39-s + 1.56·41-s + 1.82·43-s − 0.149·45-s + 1.16·47-s − 0.280·51-s + 0.269·55-s − 0.264·57-s + 1.04·59-s + 0.256·61-s + 0.496·65-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.850313331\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.850313331\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−8.836199545630867645438721125526, −7.82067649776875631788978756567, −7.51325888882796800062561076037, −6.62104364944396067563104383367, −5.67017038367597175775538292471, −4.60737339403970738856758018331, −4.18636216199252309981165234200, −2.80785568081983867387706785012, −2.44981952348987740648309403533, −0.798786744184690487080656808366,
0.798786744184690487080656808366, 2.44981952348987740648309403533, 2.80785568081983867387706785012, 4.18636216199252309981165234200, 4.60737339403970738856758018331, 5.67017038367597175775538292471, 6.62104364944396067563104383367, 7.51325888882796800062561076037, 7.82067649776875631788978756567, 8.836199545630867645438721125526
