| L(s) = 1 | + 2·3-s + 2·7-s + 9-s + 13-s − 5·17-s + 6·19-s + 4·21-s + 2·23-s − 4·27-s − 9·29-s + 2·31-s + 3·37-s + 2·39-s + 5·41-s + 2·47-s − 3·49-s − 10·51-s − 9·53-s + 12·57-s − 8·59-s − 6·61-s + 2·63-s + 2·67-s + 4·69-s − 12·71-s − 2·73-s − 10·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.755·7-s + 1/3·9-s + 0.277·13-s − 1.21·17-s + 1.37·19-s + 0.872·21-s + 0.417·23-s − 0.769·27-s − 1.67·29-s + 0.359·31-s + 0.493·37-s + 0.320·39-s + 0.780·41-s + 0.291·47-s − 3/7·49-s − 1.40·51-s − 1.23·53-s + 1.58·57-s − 1.04·59-s − 0.768·61-s + 0.251·63-s + 0.244·67-s + 0.481·69-s − 1.42·71-s − 0.234·73-s − 1.12·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.76161486742320, −14.21860758709390, −13.96763738492995, −13.26736160938200, −13.06253973171579, −12.32493131934463, −11.49153604410187, −11.28434122394773, −10.84653282440465, −9.920999793345352, −9.505605915111059, −8.934269335197531, −8.657490661685145, −7.953663372329217, −7.450447049182686, −7.206971360216451, −6.107363505517408, −5.798362330642657, −4.842132620046238, −4.503023299629143, −3.673886518185815, −3.154572763604343, −2.531006884607414, −1.831109951630538, −1.242948321528857, 0,
1.242948321528857, 1.831109951630538, 2.531006884607414, 3.154572763604343, 3.673886518185815, 4.503023299629143, 4.842132620046238, 5.798362330642657, 6.107363505517408, 7.206971360216451, 7.450447049182686, 7.953663372329217, 8.657490661685145, 8.934269335197531, 9.505605915111059, 9.920999793345352, 10.84653282440465, 11.28434122394773, 11.49153604410187, 12.32493131934463, 13.06253973171579, 13.26736160938200, 13.96763738492995, 14.21860758709390, 14.76161486742320
