| L(s) = 1 | + 4·5-s + 7-s − 5·11-s − 6·17-s − 6·19-s − 5·23-s + 11·25-s + 5·29-s − 5·31-s + 4·35-s + 9·37-s + 6·41-s − 7·43-s + 4·47-s − 6·49-s − 9·53-s − 20·55-s + 5·59-s − 2·61-s + 4·67-s − 2·71-s − 14·73-s − 5·77-s − 2·79-s − 9·83-s − 24·85-s − 3·89-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.377·7-s − 1.50·11-s − 1.45·17-s − 1.37·19-s − 1.04·23-s + 11/5·25-s + 0.928·29-s − 0.898·31-s + 0.676·35-s + 1.47·37-s + 0.937·41-s − 1.06·43-s + 0.583·47-s − 6/7·49-s − 1.23·53-s − 2.69·55-s + 0.650·59-s − 0.256·61-s + 0.488·67-s − 0.237·71-s − 1.63·73-s − 0.569·77-s − 0.225·79-s − 0.987·83-s − 2.60·85-s − 0.317·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.106329723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.106329723\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.08154895514924, −13.39862610066893, −13.23636488440076, −12.76620705398179, −12.37786363715288, −11.23853152135451, −11.13411775871561, −10.45511955420694, −10.11109235301714, −9.661827262387188, −9.020039171731878, −8.504784501874435, −8.117281960299187, −7.375236047511266, −6.707424779403563, −6.170971699791129, −5.851076682397255, −5.239869404436433, −4.553758852261400, −4.321987440531001, −3.059763259029609, −2.527302102194645, −2.056526810189466, −1.630596070153520, −0.4369172353928827,
0.4369172353928827, 1.630596070153520, 2.056526810189466, 2.527302102194645, 3.059763259029609, 4.321987440531001, 4.553758852261400, 5.239869404436433, 5.851076682397255, 6.170971699791129, 6.707424779403563, 7.375236047511266, 8.117281960299187, 8.504784501874435, 9.020039171731878, 9.661827262387188, 10.11109235301714, 10.45511955420694, 11.13411775871561, 11.23853152135451, 12.37786363715288, 12.76620705398179, 13.23636488440076, 13.39862610066893, 14.08154895514924
