| L(s) = 1 | − 3-s − 4-s + 2·7-s + 9-s + 12-s + 16-s − 2·21-s − 27-s − 2·28-s − 36-s − 37-s − 48-s + 3·49-s + 2·63-s − 64-s + 2·67-s + 2·73-s + 81-s + 2·84-s + 108-s + 111-s + 2·112-s + ⋯ |
| L(s) = 1 | − 3-s − 4-s + 2·7-s + 9-s + 12-s + 16-s − 2·21-s − 27-s − 2·28-s − 36-s − 37-s − 48-s + 3·49-s + 2·63-s − 64-s + 2·67-s + 2·73-s + 81-s + 2·84-s + 108-s + 111-s + 2·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8971150052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8971150052\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + T^{2} \) |
| 7 | \( ( 1 - T )^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−8.917730885588361634310213152757, −8.182726849364310363327924862614, −7.64505441409220898962617503230, −6.69191004372161803368460893876, −5.53344339107503011860785710335, −5.14830107742549055359894698580, −4.50153341379051409557978707378, −3.76429841424074150506165241378, −1.98462927698388361148331984147, −0.993465034244451529171560935156,
0.993465034244451529171560935156, 1.98462927698388361148331984147, 3.76429841424074150506165241378, 4.50153341379051409557978707378, 5.14830107742549055359894698580, 5.53344339107503011860785710335, 6.69191004372161803368460893876, 7.64505441409220898962617503230, 8.182726849364310363327924862614, 8.917730885588361634310213152757
