| L(s) = 1 | − 5·5-s − 14.4·7-s + 13.9·11-s − 38.8·13-s − 126.·17-s + 60.5·19-s + 50.3·23-s + 25·25-s − 71.3·29-s − 28.7·31-s + 72.4·35-s + 76.6·37-s + 396.·41-s − 351.·43-s − 4.16·47-s − 133.·49-s + 171.·53-s − 69.7·55-s + 116.·59-s + 523.·61-s + 194.·65-s − 872.·67-s + 802.·71-s − 13.5·73-s − 201.·77-s + 597.·79-s + 75.9·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.781·7-s + 0.382·11-s − 0.828·13-s − 1.79·17-s + 0.731·19-s + 0.456·23-s + 0.200·25-s − 0.456·29-s − 0.166·31-s + 0.349·35-s + 0.340·37-s + 1.50·41-s − 1.24·43-s − 0.0129·47-s − 0.388·49-s + 0.445·53-s − 0.170·55-s + 0.257·59-s + 1.09·61-s + 0.370·65-s − 1.59·67-s + 1.34·71-s − 0.0216·73-s − 0.298·77-s + 0.851·79-s + 0.100·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.169492942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.169492942\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 + 14.4T + 343T^{2} \) |
| 11 | \( 1 - 13.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 38.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 126.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 60.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 50.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 71.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 28.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 76.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 396.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 351.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 4.16T + 1.03e5T^{2} \) |
| 53 | \( 1 - 171.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 116.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 523.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 872.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 802.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 13.5T + 3.89e5T^{2} \) |
| 79 | \( 1 - 597.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 75.9T + 5.71e5T^{2} \) |
| 89 | \( 1 - 799.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 445.T + 9.12e5T^{2} \) |
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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
−9.390169818893859284987885855378, −8.865275152032340776625911800406, −7.73444377317747625784313189391, −6.96763906413348885763739451779, −6.29185566504135210752834278145, −5.09757564732186598933390165989, −4.21435411314212050572730509551, −3.22063804103471025239991673774, −2.16498507685419549847179997503, −0.54481606115123228595255219766,
0.54481606115123228595255219766, 2.16498507685419549847179997503, 3.22063804103471025239991673774, 4.21435411314212050572730509551, 5.09757564732186598933390165989, 6.29185566504135210752834278145, 6.96763906413348885763739451779, 7.73444377317747625784313189391, 8.865275152032340776625911800406, 9.390169818893859284987885855378
