| L(s) = 1 | − 2.04·3-s + 4.25·5-s − 7-s + 1.17·9-s − 11-s + 13-s − 8.69·15-s − 2.35·17-s + 5.32·19-s + 2.04·21-s + 3.35·23-s + 13.1·25-s + 3.72·27-s − 0.720·29-s − 9.19·31-s + 2.04·33-s − 4.25·35-s + 7.52·37-s − 2.04·39-s + 1.34·41-s − 5.57·43-s + 5.01·45-s − 5.81·47-s + 49-s + 4.80·51-s + 3.10·53-s − 4.25·55-s + ⋯ |
| L(s) = 1 | − 1.18·3-s + 1.90·5-s − 0.377·7-s + 0.392·9-s − 0.301·11-s + 0.277·13-s − 2.24·15-s − 0.570·17-s + 1.22·19-s + 0.446·21-s + 0.698·23-s + 2.62·25-s + 0.716·27-s − 0.133·29-s − 1.65·31-s + 0.355·33-s − 0.719·35-s + 1.23·37-s − 0.327·39-s + 0.210·41-s − 0.849·43-s + 0.747·45-s − 0.848·47-s + 0.142·49-s + 0.673·51-s + 0.426·53-s − 0.573·55-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
| \(L(1)\) |
\(\approx\) |
\(1.816610907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.816610907\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.04T + 3T^{2} \) |
| 5 | \( 1 - 4.25T + 5T^{2} \) |
| 17 | \( 1 + 2.35T + 17T^{2} \) |
| 19 | \( 1 - 5.32T + 19T^{2} \) |
| 23 | \( 1 - 3.35T + 23T^{2} \) |
| 29 | \( 1 + 0.720T + 29T^{2} \) |
| 31 | \( 1 + 9.19T + 31T^{2} \) |
| 37 | \( 1 - 7.52T + 37T^{2} \) |
| 41 | \( 1 - 1.34T + 41T^{2} \) |
| 43 | \( 1 + 5.57T + 43T^{2} \) |
| 47 | \( 1 + 5.81T + 47T^{2} \) |
| 53 | \( 1 - 3.10T + 53T^{2} \) |
| 59 | \( 1 - 2.11T + 59T^{2} \) |
| 61 | \( 1 - 9.70T + 61T^{2} \) |
| 67 | \( 1 + 5.48T + 67T^{2} \) |
| 71 | \( 1 + 7.12T + 71T^{2} \) |
| 73 | \( 1 + 0.258T + 73T^{2} \) |
| 79 | \( 1 - 6.57T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 3.05T + 89T^{2} \) |
| 97 | \( 1 + 6.67T + 97T^{2} \) |
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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
−7.61991776435620420526070421562, −6.78998558793260226293452520353, −6.36529488741822596942408578256, −5.64783848610910990924274363687, −5.34841223230445404193295400329, −4.68931723141976878460196609112, −3.38099816571966522319071291996, −2.55770136223978149241546960984, −1.65867740225732509729629147428, −0.72906398370194579746092105447,
0.72906398370194579746092105447, 1.65867740225732509729629147428, 2.55770136223978149241546960984, 3.38099816571966522319071291996, 4.68931723141976878460196609112, 5.34841223230445404193295400329, 5.64783848610910990924274363687, 6.36529488741822596942408578256, 6.78998558793260226293452520353, 7.61991776435620420526070421562
