| L(s) = 1 | − 0.618·3-s + 1.61·5-s − 7-s − 2.61·9-s + 11-s − 13-s − 1.00·15-s − 5.85·17-s − 4.09·19-s + 0.618·21-s − 6.47·23-s − 2.38·25-s + 3.47·27-s − 0.618·33-s − 1.61·35-s + 7.23·37-s + 0.618·39-s + 1.70·41-s − 7.09·43-s − 4.23·45-s − 4·47-s + 49-s + 3.61·51-s + 5.38·53-s + 1.61·55-s + 2.52·57-s + 6·59-s + ⋯ |
| L(s) = 1 | − 0.356·3-s + 0.723·5-s − 0.377·7-s − 0.872·9-s + 0.301·11-s − 0.277·13-s − 0.258·15-s − 1.41·17-s − 0.938·19-s + 0.134·21-s − 1.34·23-s − 0.476·25-s + 0.668·27-s − 0.107·33-s − 0.273·35-s + 1.18·37-s + 0.0989·39-s + 0.266·41-s − 1.08·43-s − 0.631·45-s − 0.583·47-s + 0.142·49-s + 0.506·51-s + 0.739·53-s + 0.218·55-s + 0.334·57-s + 0.781·59-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
| \(L(1)\) |
\(\approx\) |
\(1.081513654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.081513654\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.618T + 3T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 17 | \( 1 + 5.85T + 17T^{2} \) |
| 19 | \( 1 + 4.09T + 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 7.23T + 37T^{2} \) |
| 41 | \( 1 - 1.70T + 41T^{2} \) |
| 43 | \( 1 + 7.09T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 5.38T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 4.14T + 61T^{2} \) |
| 67 | \( 1 - 9.38T + 67T^{2} \) |
| 71 | \( 1 - 1.90T + 71T^{2} \) |
| 73 | \( 1 - 7.70T + 73T^{2} \) |
| 79 | \( 1 - 2.38T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 7.85T + 89T^{2} \) |
| 97 | \( 1 - 2.94T + 97T^{2} \) |
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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
−7.962012910633720953218213045880, −6.86845906974759804657973504486, −6.33579525590797103410232898866, −5.94011693254101691768791419811, −5.12496525060139994173417368556, −4.33057952175874938587359541844, −3.56318852774364704102060607471, −2.38807905729854354043968971114, −2.04566338285766587448203693664, −0.49213820577522629710346314376,
0.49213820577522629710346314376, 2.04566338285766587448203693664, 2.38807905729854354043968971114, 3.56318852774364704102060607471, 4.33057952175874938587359541844, 5.12496525060139994173417368556, 5.94011693254101691768791419811, 6.33579525590797103410232898866, 6.86845906974759804657973504486, 7.962012910633720953218213045880
