| L(s) = 1 | − 3-s − 2·9-s − 3·11-s − 2·13-s − 3·17-s − 19-s − 3·23-s + 5·27-s − 6·29-s − 7·31-s + 3·33-s + 37-s + 2·39-s + 6·41-s + 4·43-s + 9·47-s + 3·51-s − 3·53-s + 57-s + 9·59-s − 61-s + 7·67-s + 3·69-s + 73-s − 13·79-s + 81-s − 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s − 0.904·11-s − 0.554·13-s − 0.727·17-s − 0.229·19-s − 0.625·23-s + 0.962·27-s − 1.11·29-s − 1.25·31-s + 0.522·33-s + 0.164·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s + 1.31·47-s + 0.420·51-s − 0.412·53-s + 0.132·57-s + 1.17·59-s − 0.128·61-s + 0.855·67-s + 0.361·69-s + 0.117·73-s − 1.46·79-s + 1/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7009066754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7009066754\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.206314383565699032583413773277, −7.50583850856640237230837436037, −6.85420158442897404526323646241, −5.81685939880161480103979342246, −5.56063789989129311990774760162, −4.65518649907072722746944298509, −3.83503074284329452660005843394, −2.72320977868378818164957546956, −2.04516451501611384471824498227, −0.44464023562677003765543182427,
0.44464023562677003765543182427, 2.04516451501611384471824498227, 2.72320977868378818164957546956, 3.83503074284329452660005843394, 4.65518649907072722746944298509, 5.56063789989129311990774760162, 5.81685939880161480103979342246, 6.85420158442897404526323646241, 7.50583850856640237230837436037, 8.206314383565699032583413773277
