L(s) = 1 | − 5-s + 7-s − 2·13-s − 2·17-s + 7·19-s + 6·23-s + 25-s − 8·29-s + 3·31-s − 35-s + 37-s + 6·41-s − 4·47-s − 6·49-s − 10·53-s + 4·59-s − 11·61-s + 2·65-s + 5·67-s + 6·71-s + 73-s + 79-s − 12·83-s + 2·85-s − 10·89-s − 2·91-s − 7·95-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s − 0.554·13-s − 0.485·17-s + 1.60·19-s + 1.25·23-s + 1/5·25-s − 1.48·29-s + 0.538·31-s − 0.169·35-s + 0.164·37-s + 0.937·41-s − 0.583·47-s − 6/7·49-s − 1.37·53-s + 0.520·59-s − 1.40·61-s + 0.248·65-s + 0.610·67-s + 0.712·71-s + 0.117·73-s + 0.112·79-s − 1.31·83-s + 0.216·85-s − 1.05·89-s − 0.209·91-s − 0.718·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−14.31979633016342, −13.64993376187751, −13.12181197598779, −12.65154943198193, −12.23630086937228, −11.47416037802681, −11.22541944379962, −10.98010028830194, −10.01617467141910, −9.680457474373905, −9.167755733256785, −8.669317132989238, −7.962094960294143, −7.553100523934415, −7.192071741817243, −6.559281369821055, −5.880859254318238, −5.261752047930851, −4.801303997911153, −4.345781133524036, −3.459255210052786, −3.116970739963857, −2.371759992867843, −1.578850407892284, −0.9088450905320447, 0,
0.9088450905320447, 1.578850407892284, 2.371759992867843, 3.116970739963857, 3.459255210052786, 4.345781133524036, 4.801303997911153, 5.261752047930851, 5.880859254318238, 6.559281369821055, 7.192071741817243, 7.553100523934415, 7.962094960294143, 8.669317132989238, 9.167755733256785, 9.680457474373905, 10.01617467141910, 10.98010028830194, 11.22541944379962, 11.47416037802681, 12.23630086937228, 12.65154943198193, 13.12181197598779, 13.64993376187751, 14.31979633016342