| L(s) = 1 | + 5·5-s + 7·7-s + 9·9-s + 6·17-s + 18·19-s + 25·25-s + 35·35-s − 66·37-s + 54·43-s + 45·45-s − 66·47-s + 49·49-s − 34·53-s − 62·59-s − 102·61-s + 63·63-s + 6·67-s + 138·71-s − 106·73-s + 122·79-s + 81·81-s + 30·85-s + 90·95-s + 166·97-s − 22·101-s + 46·103-s − 74·107-s + ⋯ |
| L(s) = 1 | + 5-s + 7-s + 9-s + 6/17·17-s + 0.947·19-s + 25-s + 35-s − 1.78·37-s + 1.25·43-s + 45-s − 1.40·47-s + 49-s − 0.641·53-s − 1.05·59-s − 1.67·61-s + 63-s + 6/67·67-s + 1.94·71-s − 1.45·73-s + 1.54·79-s + 81-s + 6/17·85-s + 0.947·95-s + 1.71·97-s − 0.217·101-s + 0.446·103-s − 0.691·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.047716580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.047716580\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( 1 - 6 T + p^{2} T^{2} \) |
| 19 | \( 1 - 18 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 + 66 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 - 54 T + p^{2} T^{2} \) |
| 47 | \( 1 + 66 T + p^{2} T^{2} \) |
| 53 | \( 1 + 34 T + p^{2} T^{2} \) |
| 59 | \( 1 + 62 T + p^{2} T^{2} \) |
| 61 | \( 1 + 102 T + p^{2} T^{2} \) |
| 67 | \( 1 - 6 T + p^{2} T^{2} \) |
| 71 | \( 1 - 138 T + p^{2} T^{2} \) |
| 73 | \( 1 + 106 T + p^{2} T^{2} \) |
| 79 | \( 1 - 122 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 - 166 T + p^{2} T^{2} \) |
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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
−9.651442294510114882630310402944, −8.964996463131828389506616895733, −7.905535103184630417738006812801, −7.21104226824344991818633619393, −6.24266375419050125531190026118, −5.25268553096086458033284675916, −4.64784410779914454672524440361, −3.34790966803959958112130083263, −1.98348170876347941748888704469, −1.20891156481929634183234679436,
1.20891156481929634183234679436, 1.98348170876347941748888704469, 3.34790966803959958112130083263, 4.64784410779914454672524440361, 5.25268553096086458033284675916, 6.24266375419050125531190026118, 7.21104226824344991818633619393, 7.905535103184630417738006812801, 8.964996463131828389506616895733, 9.651442294510114882630310402944
