| L(s) = 1 | − 0.765·3-s − 1.84·5-s − 7-s − 2.41·9-s − 3.69·11-s − 2.29·13-s + 1.41·15-s + 0.828·17-s − 3.37·19-s + 0.765·21-s + 0.585·23-s − 1.58·25-s + 4.14·27-s − 8.92·29-s + 1.17·31-s + 2.82·33-s + 1.84·35-s + 0.634·37-s + 1.75·39-s − 6.48·41-s − 1.53·43-s + 4.46·45-s + 4·47-s + 49-s − 0.634·51-s + 7.39·53-s + 6.82·55-s + ⋯ |
| L(s) = 1 | − 0.441·3-s − 0.826·5-s − 0.377·7-s − 0.804·9-s − 1.11·11-s − 0.636·13-s + 0.365·15-s + 0.200·17-s − 0.775·19-s + 0.167·21-s + 0.122·23-s − 0.317·25-s + 0.797·27-s − 1.65·29-s + 0.210·31-s + 0.492·33-s + 0.312·35-s + 0.104·37-s + 0.281·39-s − 1.01·41-s − 0.233·43-s + 0.664·45-s + 0.583·47-s + 0.142·49-s − 0.0887·51-s + 1.01·53-s + 0.920·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3552691665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3552691665\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 0.765T + 3T^{2} \) |
| 5 | \( 1 + 1.84T + 5T^{2} \) |
| 11 | \( 1 + 3.69T + 11T^{2} \) |
| 13 | \( 1 + 2.29T + 13T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 19 | \( 1 + 3.37T + 19T^{2} \) |
| 23 | \( 1 - 0.585T + 23T^{2} \) |
| 29 | \( 1 + 8.92T + 29T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 - 0.634T + 37T^{2} \) |
| 41 | \( 1 + 6.48T + 41T^{2} \) |
| 43 | \( 1 + 1.53T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 7.39T + 53T^{2} \) |
| 59 | \( 1 + 15.0T + 59T^{2} \) |
| 61 | \( 1 - 1.84T + 61T^{2} \) |
| 67 | \( 1 - 6.49T + 67T^{2} \) |
| 71 | \( 1 - 4.82T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 8.82T + 79T^{2} \) |
| 83 | \( 1 - 2.29T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.435896856223279955526004057589, −7.78855044741218568231544659084, −7.19817469258279948667174274915, −6.24654092239250354599854360729, −5.51725979299595917448592402412, −4.86333128488602900128200134918, −3.87922650391284749474284875675, −3.04640810099335768491267406323, −2.13737627105708479104635022905, −0.33216338296864354427139536722,
0.33216338296864354427139536722, 2.13737627105708479104635022905, 3.04640810099335768491267406323, 3.87922650391284749474284875675, 4.86333128488602900128200134918, 5.51725979299595917448592402412, 6.24654092239250354599854360729, 7.19817469258279948667174274915, 7.78855044741218568231544659084, 8.435896856223279955526004057589
