| L(s) = 1 | + 5-s − 3.60·7-s + 1.60·11-s + 0.640·13-s − 5.28·17-s + 19-s + 8.24·23-s + 25-s + 4.88·29-s − 7.28·31-s − 3.60·35-s − 9.92·37-s − 4.57·41-s + 7.92·43-s − 0.249·47-s + 6.03·49-s − 1.75·53-s + 1.60·55-s − 12.3·59-s + 11.5·61-s + 0.640·65-s + 12.4·67-s − 13.7·71-s + 3.93·73-s − 5.81·77-s − 3.03·79-s + 10.4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.36·7-s + 0.485·11-s + 0.177·13-s − 1.28·17-s + 0.229·19-s + 1.72·23-s + 0.200·25-s + 0.908·29-s − 1.30·31-s − 0.610·35-s − 1.63·37-s − 0.715·41-s + 1.20·43-s − 0.0364·47-s + 0.861·49-s − 0.240·53-s + 0.217·55-s − 1.60·59-s + 1.47·61-s + 0.0793·65-s + 1.52·67-s − 1.63·71-s + 0.461·73-s − 0.662·77-s − 0.340·79-s + 1.15·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 - 1.60T + 11T^{2} \) |
| 13 | \( 1 - 0.640T + 13T^{2} \) |
| 17 | \( 1 + 5.28T + 17T^{2} \) |
| 23 | \( 1 - 8.24T + 23T^{2} \) |
| 29 | \( 1 - 4.88T + 29T^{2} \) |
| 31 | \( 1 + 7.28T + 31T^{2} \) |
| 37 | \( 1 + 9.92T + 37T^{2} \) |
| 41 | \( 1 + 4.57T + 41T^{2} \) |
| 43 | \( 1 - 7.92T + 43T^{2} \) |
| 47 | \( 1 + 0.249T + 47T^{2} \) |
| 53 | \( 1 + 1.75T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 3.93T + 73T^{2} \) |
| 79 | \( 1 + 3.03T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 + 9.92T + 97T^{2} \) |
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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
−7.35386502901359475023084460706, −6.77748491569692555397128297198, −6.42585845987794153338652239304, −5.53774723430846712494652222077, −4.82484496940987431037165665058, −3.84068368974082680258838694489, −3.17654747000103643595513570193, −2.38829477115512194130026529767, −1.27098194994325741981027740872, 0,
1.27098194994325741981027740872, 2.38829477115512194130026529767, 3.17654747000103643595513570193, 3.84068368974082680258838694489, 4.82484496940987431037165665058, 5.53774723430846712494652222077, 6.42585845987794153338652239304, 6.77748491569692555397128297198, 7.35386502901359475023084460706
