| L(s) = 1 | − 1.36·3-s − 3.92·5-s + 7-s − 1.12·9-s − 11-s − 13-s + 5.37·15-s − 2.95·17-s − 3.52·19-s − 1.36·21-s + 4.25·23-s + 10.3·25-s + 5.64·27-s − 9.21·29-s + 1.84·31-s + 1.36·33-s − 3.92·35-s + 0.471·37-s + 1.36·39-s + 6.70·41-s + 12.6·43-s + 4.41·45-s + 5.58·47-s + 49-s + 4.05·51-s + 12.0·53-s + 3.92·55-s + ⋯ |
| L(s) = 1 | − 0.790·3-s − 1.75·5-s + 0.377·7-s − 0.375·9-s − 0.301·11-s − 0.277·13-s + 1.38·15-s − 0.717·17-s − 0.807·19-s − 0.298·21-s + 0.886·23-s + 2.07·25-s + 1.08·27-s − 1.71·29-s + 0.332·31-s + 0.238·33-s − 0.663·35-s + 0.0775·37-s + 0.219·39-s + 1.04·41-s + 1.92·43-s + 0.658·45-s + 0.814·47-s + 0.142·49-s + 0.567·51-s + 1.66·53-s + 0.529·55-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.36T + 3T^{2} \) |
| 5 | \( 1 + 3.92T + 5T^{2} \) |
| 17 | \( 1 + 2.95T + 17T^{2} \) |
| 19 | \( 1 + 3.52T + 19T^{2} \) |
| 23 | \( 1 - 4.25T + 23T^{2} \) |
| 29 | \( 1 + 9.21T + 29T^{2} \) |
| 31 | \( 1 - 1.84T + 31T^{2} \) |
| 37 | \( 1 - 0.471T + 37T^{2} \) |
| 41 | \( 1 - 6.70T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 - 5.58T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 + 7.88T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 0.782T + 67T^{2} \) |
| 71 | \( 1 + 0.831T + 71T^{2} \) |
| 73 | \( 1 + 7.35T + 73T^{2} \) |
| 79 | \( 1 - 3.31T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 9.73T + 89T^{2} \) |
| 97 | \( 1 + 0.699T + 97T^{2} \) |
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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
−7.49569837956531194722615215422, −6.96182303945641722583459679256, −6.03332955466003676439600478493, −5.38960126743585175673723370337, −4.46127911946999236177084813916, −4.21847233972453388057733785509, −3.17597622235406965217435100405, −2.33222925126828449600265891159, −0.824736239424921146838993793356, 0,
0.824736239424921146838993793356, 2.33222925126828449600265891159, 3.17597622235406965217435100405, 4.21847233972453388057733785509, 4.46127911946999236177084813916, 5.38960126743585175673723370337, 6.03332955466003676439600478493, 6.96182303945641722583459679256, 7.49569837956531194722615215422
