| L(s) = 1 | − 2.25·3-s − 2.11·5-s + 7-s + 2.10·9-s − 11-s + 13-s + 4.78·15-s − 0.741·17-s − 0.409·19-s − 2.25·21-s − 3.88·23-s − 0.508·25-s + 2.02·27-s + 2.62·29-s − 1.76·31-s + 2.25·33-s − 2.11·35-s − 6.82·37-s − 2.25·39-s − 6.25·41-s − 1.02·43-s − 4.45·45-s + 12.1·47-s + 49-s + 1.67·51-s + 9.93·53-s + 2.11·55-s + ⋯ |
| L(s) = 1 | − 1.30·3-s − 0.947·5-s + 0.377·7-s + 0.700·9-s − 0.301·11-s + 0.277·13-s + 1.23·15-s − 0.179·17-s − 0.0938·19-s − 0.492·21-s − 0.811·23-s − 0.101·25-s + 0.390·27-s + 0.486·29-s − 0.316·31-s + 0.393·33-s − 0.358·35-s − 1.12·37-s − 0.361·39-s − 0.977·41-s − 0.155·43-s − 0.664·45-s + 1.77·47-s + 0.142·49-s + 0.234·51-s + 1.36·53-s + 0.285·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5155401603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5155401603\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2.25T + 3T^{2} \) |
| 5 | \( 1 + 2.11T + 5T^{2} \) |
| 17 | \( 1 + 0.741T + 17T^{2} \) |
| 19 | \( 1 + 0.409T + 19T^{2} \) |
| 23 | \( 1 + 3.88T + 23T^{2} \) |
| 29 | \( 1 - 2.62T + 29T^{2} \) |
| 31 | \( 1 + 1.76T + 31T^{2} \) |
| 37 | \( 1 + 6.82T + 37T^{2} \) |
| 41 | \( 1 + 6.25T + 41T^{2} \) |
| 43 | \( 1 + 1.02T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 - 9.93T + 53T^{2} \) |
| 59 | \( 1 + 4.94T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 - 0.521T + 67T^{2} \) |
| 71 | \( 1 + 9.18T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 - 1.19T + 79T^{2} \) |
| 83 | \( 1 + 2.13T + 83T^{2} \) |
| 89 | \( 1 - 8.71T + 89T^{2} \) |
| 97 | \( 1 + 9.83T + 97T^{2} \) |
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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
−7.72263013687954817746887655480, −7.09980603438224850335399985118, −6.43093525819029359728234930422, −5.63264610712819978278195444939, −5.18701100319707904124703578234, −4.29657736891703835512251622634, −3.81571931437529057146765045482, −2.68412938571477414687433634533, −1.51578274092595624927880955310, −0.39139964977841404939409304024,
0.39139964977841404939409304024, 1.51578274092595624927880955310, 2.68412938571477414687433634533, 3.81571931437529057146765045482, 4.29657736891703835512251622634, 5.18701100319707904124703578234, 5.63264610712819978278195444939, 6.43093525819029359728234930422, 7.09980603438224850335399985118, 7.72263013687954817746887655480
