| L(s) = 1 | + 5-s − 0.732·7-s + 11-s − 6.19·13-s − 1.26·17-s + 0.535·19-s + 6.92·23-s + 25-s + 6.92·29-s + 7.46·31-s − 0.732·35-s − 7.46·37-s − 6.92·41-s + 1.80·43-s + 2.53·47-s − 6.46·49-s + 6·53-s + 55-s − 12.9·59-s − 0.535·61-s − 6.19·65-s + 4·67-s − 6·71-s + 7.66·73-s − 0.732·77-s + 0.535·79-s + 1.26·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.276·7-s + 0.301·11-s − 1.71·13-s − 0.307·17-s + 0.122·19-s + 1.44·23-s + 0.200·25-s + 1.28·29-s + 1.34·31-s − 0.123·35-s − 1.22·37-s − 1.08·41-s + 0.275·43-s + 0.369·47-s − 0.923·49-s + 0.824·53-s + 0.134·55-s − 1.68·59-s − 0.0686·61-s − 0.768·65-s + 0.488·67-s − 0.712·71-s + 0.896·73-s − 0.0834·77-s + 0.0602·79-s + 0.139·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.884610667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.884610667\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 + 0.732T + 7T^{2} \) |
| 13 | \( 1 + 6.19T + 13T^{2} \) |
| 17 | \( 1 + 1.26T + 17T^{2} \) |
| 19 | \( 1 - 0.535T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 - 7.46T + 31T^{2} \) |
| 37 | \( 1 + 7.46T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 - 1.80T + 43T^{2} \) |
| 47 | \( 1 - 2.53T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 0.535T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 7.66T + 73T^{2} \) |
| 79 | \( 1 - 0.535T + 79T^{2} \) |
| 83 | \( 1 - 1.26T + 83T^{2} \) |
| 89 | \( 1 + 0.928T + 89T^{2} \) |
| 97 | \( 1 - 8.92T + 97T^{2} \) |
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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
−7.79539519753405424820793261543, −6.91945531586447717754995937070, −6.70801507287269089098943214605, −5.74633752749334488545140369189, −4.84158599893694269745135142915, −4.63647511576311340854072330659, −3.29240384886517595945615391512, −2.73937053468560504021982610332, −1.84087159187009644490714231449, −0.67292998215298395859127371294,
0.67292998215298395859127371294, 1.84087159187009644490714231449, 2.73937053468560504021982610332, 3.29240384886517595945615391512, 4.63647511576311340854072330659, 4.84158599893694269745135142915, 5.74633752749334488545140369189, 6.70801507287269089098943214605, 6.91945531586447717754995937070, 7.79539519753405424820793261543
