| L(s) = 1 | + 3-s − 3.41·5-s + 9-s − 0.828·11-s + 4.24·13-s − 3.41·15-s + 7.41·17-s − 6.82·19-s − 4.82·23-s + 6.65·25-s + 27-s − 2.82·29-s − 2.82·31-s − 0.828·33-s + 1.65·37-s + 4.24·39-s + 10.2·41-s + 11.3·43-s − 3.41·45-s − 4.48·47-s + 7.41·51-s + 2·53-s + 2.82·55-s − 6.82·57-s − 8.48·59-s − 11.0·61-s − 14.4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.52·5-s + 0.333·9-s − 0.249·11-s + 1.17·13-s − 0.881·15-s + 1.79·17-s − 1.56·19-s − 1.00·23-s + 1.33·25-s + 0.192·27-s − 0.525·29-s − 0.508·31-s − 0.144·33-s + 0.272·37-s + 0.679·39-s + 1.59·41-s + 1.72·43-s − 0.508·45-s − 0.654·47-s + 1.03·51-s + 0.274·53-s + 0.381·55-s − 0.904·57-s − 1.10·59-s − 1.41·61-s − 1.79·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.700465382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.700465382\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3.41T + 5T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 7.41T + 17T^{2} \) |
| 19 | \( 1 + 6.82T + 19T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 - 1.65T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 7.75T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 5.75T + 89T^{2} \) |
| 97 | \( 1 + 0.242T + 97T^{2} \) |
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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
−7.72275008542944268735737916750, −7.44808884543647556905944138475, −6.24760848126354811361698208045, −5.83749392489465073799054042500, −4.64463148681494043270224341480, −4.01135216538459630281032923746, −3.59402212957767371259867158690, −2.82219856392966073657043326428, −1.70652957926291254770572958719, −0.61369092890532234282127878369,
0.61369092890532234282127878369, 1.70652957926291254770572958719, 2.82219856392966073657043326428, 3.59402212957767371259867158690, 4.01135216538459630281032923746, 4.64463148681494043270224341480, 5.83749392489465073799054042500, 6.24760848126354811361698208045, 7.44808884543647556905944138475, 7.72275008542944268735737916750
