| L(s) = 1 | + 5-s − 5·11-s − 2·13-s − 6·17-s + 2·19-s + 6·23-s − 4·25-s − 3·29-s + 5·31-s − 2·37-s − 8·41-s + 4·43-s + 4·47-s + 9·53-s − 5·55-s − 3·59-s + 12·61-s − 2·65-s − 2·67-s + 8·71-s + 14·73-s − 79-s + 17·83-s − 6·85-s − 18·89-s + 2·95-s − 3·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.50·11-s − 0.554·13-s − 1.45·17-s + 0.458·19-s + 1.25·23-s − 4/5·25-s − 0.557·29-s + 0.898·31-s − 0.328·37-s − 1.24·41-s + 0.609·43-s + 0.583·47-s + 1.23·53-s − 0.674·55-s − 0.390·59-s + 1.53·61-s − 0.248·65-s − 0.244·67-s + 0.949·71-s + 1.63·73-s − 0.112·79-s + 1.86·83-s − 0.650·85-s − 1.90·89-s + 0.205·95-s − 0.304·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.516542519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.516542519\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−7.940963875965950811187412500021, −7.18647837334672438523571766437, −6.66813384608020900552232584451, −5.68098313333067741075620523599, −5.16194350111874490179185123126, −4.53384136488930654225510718868, −3.48029655211268921609097017103, −2.53384233463331346727195600031, −2.06638425598003929422411599917, −0.59569207132295766078901924349,
0.59569207132295766078901924349, 2.06638425598003929422411599917, 2.53384233463331346727195600031, 3.48029655211268921609097017103, 4.53384136488930654225510718868, 5.16194350111874490179185123126, 5.68098313333067741075620523599, 6.66813384608020900552232584451, 7.18647837334672438523571766437, 7.940963875965950811187412500021
