| L(s) = 1 | − 3·5-s + 11-s + 13-s + 4·17-s + 5·19-s + 6·23-s + 4·25-s − 3·29-s − 2·31-s − 11·37-s + 6·43-s + 3·47-s − 8·53-s − 3·55-s − 5·59-s − 10·61-s − 3·65-s + 5·67-s + 3·73-s − 6·79-s + 16·83-s − 12·85-s − 14·89-s − 15·95-s − 18·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.301·11-s + 0.277·13-s + 0.970·17-s + 1.14·19-s + 1.25·23-s + 4/5·25-s − 0.557·29-s − 0.359·31-s − 1.80·37-s + 0.914·43-s + 0.437·47-s − 1.09·53-s − 0.404·55-s − 0.650·59-s − 1.28·61-s − 0.372·65-s + 0.610·67-s + 0.351·73-s − 0.675·79-s + 1.75·83-s − 1.30·85-s − 1.48·89-s − 1.53·95-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−15.30178965524066, −14.56133699843850, −14.07494782978862, −13.64875003094237, −12.83206151824020, −12.34602643367541, −12.01543075219693, −11.44500087179241, −10.90908826110461, −10.58420760657062, −9.613827283053934, −9.302089151637752, −8.611925497171273, −8.080040704402840, −7.407312894396771, −7.275050170526337, −6.496176563636139, −5.664770035627953, −5.177374320420507, −4.522437798926743, −3.748282716568980, −3.393655842901459, −2.820436285593137, −1.606541529307956, −0.9633665042287295, 0,
0.9633665042287295, 1.606541529307956, 2.820436285593137, 3.393655842901459, 3.748282716568980, 4.522437798926743, 5.177374320420507, 5.664770035627953, 6.496176563636139, 7.275050170526337, 7.407312894396771, 8.080040704402840, 8.611925497171273, 9.302089151637752, 9.613827283053934, 10.58420760657062, 10.90908826110461, 11.44500087179241, 12.01543075219693, 12.34602643367541, 12.83206151824020, 13.64875003094237, 14.07494782978862, 14.56133699843850, 15.30178965524066
