| L(s) = 1 | + 3·3-s − 16.8·5-s + 7.69·7-s + 9·9-s + 11·11-s + 24.8·13-s − 50.5·15-s − 15.9·17-s − 15.1·19-s + 23.0·21-s − 17.7·23-s + 158.·25-s + 27·27-s − 128.·29-s − 219.·31-s + 33·33-s − 129.·35-s + 92.0·37-s + 74.5·39-s − 459.·41-s − 64.9·43-s − 151.·45-s − 497.·47-s − 283.·49-s − 47.8·51-s − 526.·53-s − 185.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.50·5-s + 0.415·7-s + 0.333·9-s + 0.301·11-s + 0.530·13-s − 0.870·15-s − 0.227·17-s − 0.182·19-s + 0.239·21-s − 0.160·23-s + 1.27·25-s + 0.192·27-s − 0.823·29-s − 1.27·31-s + 0.174·33-s − 0.626·35-s + 0.409·37-s + 0.306·39-s − 1.75·41-s − 0.230·43-s − 0.502·45-s − 1.54·47-s − 0.827·49-s − 0.131·51-s − 1.36·53-s − 0.454·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| good | 5 | \( 1 + 16.8T + 125T^{2} \) |
| 7 | \( 1 - 7.69T + 343T^{2} \) |
| 13 | \( 1 - 24.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 15.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 15.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 17.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 128.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 219.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 92.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 459.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 64.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 497.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 526.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 578.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 221.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 860.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 580.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 510.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 606.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 23.4T + 7.04e5T^{2} \) |
| 97 | \( 1 - 719.T + 9.12e5T^{2} \) |
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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
−9.959088829180883620329534705200, −8.848569820353239873905411581597, −8.204999707146142173908727035681, −7.49956260017542225733054480131, −6.55561076875186715878678075720, −5.03664054129994538883696061922, −3.99360765766063151571840891145, −3.31715648920465848974587737425, −1.66933105027724299302279399416, 0,
1.66933105027724299302279399416, 3.31715648920465848974587737425, 3.99360765766063151571840891145, 5.03664054129994538883696061922, 6.55561076875186715878678075720, 7.49956260017542225733054480131, 8.204999707146142173908727035681, 8.848569820353239873905411581597, 9.959088829180883620329534705200
