| L(s) = 1 | − 1.78·5-s − 3.29·7-s + 1.53·11-s + 0.585·13-s − 6.08·17-s − 1.92·19-s + 5.22·23-s − 1.82·25-s − 6.81·29-s + 6.01·31-s + 5.86·35-s + 3.41·37-s − 1.04·41-s − 11.2·43-s + 0.896·47-s + 3.82·49-s + 6.81·53-s − 2.72·55-s − 10.4·59-s + 4.58·61-s − 1.04·65-s − 9.30·67-s − 14.7·71-s − 6.48·73-s − 5.03·77-s + 3.29·79-s − 13.2·83-s + ⋯ |
| L(s) = 1 | − 0.796·5-s − 1.24·7-s + 0.461·11-s + 0.162·13-s − 1.47·17-s − 0.442·19-s + 1.08·23-s − 0.365·25-s − 1.26·29-s + 1.08·31-s + 0.990·35-s + 0.561·37-s − 0.162·41-s − 1.71·43-s + 0.130·47-s + 0.546·49-s + 0.936·53-s − 0.367·55-s − 1.36·59-s + 0.587·61-s − 0.129·65-s − 1.13·67-s − 1.75·71-s − 0.759·73-s − 0.574·77-s + 0.370·79-s − 1.45·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7131270048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7131270048\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.78T + 5T^{2} \) |
| 7 | \( 1 + 3.29T + 7T^{2} \) |
| 11 | \( 1 - 1.53T + 11T^{2} \) |
| 13 | \( 1 - 0.585T + 13T^{2} \) |
| 17 | \( 1 + 6.08T + 17T^{2} \) |
| 19 | \( 1 + 1.92T + 19T^{2} \) |
| 23 | \( 1 - 5.22T + 23T^{2} \) |
| 29 | \( 1 + 6.81T + 29T^{2} \) |
| 31 | \( 1 - 6.01T + 31T^{2} \) |
| 37 | \( 1 - 3.41T + 37T^{2} \) |
| 41 | \( 1 + 1.04T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 0.896T + 47T^{2} \) |
| 53 | \( 1 - 6.81T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 4.58T + 61T^{2} \) |
| 67 | \( 1 + 9.30T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 + 6.48T + 73T^{2} \) |
| 79 | \( 1 - 3.29T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 7.12T + 89T^{2} \) |
| 97 | \( 1 - 7.31T + 97T^{2} \) |
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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
−7.53946895805986176458475818364, −7.04725623458282718028696777722, −6.38980571294807474639865421334, −5.89444775943370831698027604856, −4.73083420627562701513614961623, −4.20092731768893569193798267231, −3.43875115327743626919684874699, −2.82075625826789992970348907517, −1.73579672227852271443544395487, −0.39344218012415796689052972061,
0.39344218012415796689052972061, 1.73579672227852271443544395487, 2.82075625826789992970348907517, 3.43875115327743626919684874699, 4.20092731768893569193798267231, 4.73083420627562701513614961623, 5.89444775943370831698027604856, 6.38980571294807474639865421334, 7.04725623458282718028696777722, 7.53946895805986176458475818364
