| L(s) = 1 | − 0.477·2-s − 1.77·4-s + 1.98·5-s − 7-s + 1.79·8-s − 0.947·10-s + 0.129·11-s + 5.16·13-s + 0.477·14-s + 2.68·16-s + 2.68·17-s + 4.39·19-s − 3.52·20-s − 0.0615·22-s + 2.43·23-s − 1.05·25-s − 2.46·26-s + 1.77·28-s − 5.32·29-s + 2.06·31-s − 4.88·32-s − 1.27·34-s − 1.98·35-s + 2.29·37-s − 2.09·38-s + 3.57·40-s + 3.69·41-s + ⋯ |
| L(s) = 1 | − 0.337·2-s − 0.886·4-s + 0.888·5-s − 0.377·7-s + 0.636·8-s − 0.299·10-s + 0.0389·11-s + 1.43·13-s + 0.127·14-s + 0.671·16-s + 0.650·17-s + 1.00·19-s − 0.787·20-s − 0.0131·22-s + 0.508·23-s − 0.210·25-s − 0.483·26-s + 0.334·28-s − 0.988·29-s + 0.370·31-s − 0.862·32-s − 0.219·34-s − 0.335·35-s + 0.376·37-s − 0.340·38-s + 0.565·40-s + 0.577·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.903481355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.903481355\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 0.477T + 2T^{2} \) |
| 5 | \( 1 - 1.98T + 5T^{2} \) |
| 11 | \( 1 - 0.129T + 11T^{2} \) |
| 13 | \( 1 - 5.16T + 13T^{2} \) |
| 17 | \( 1 - 2.68T + 17T^{2} \) |
| 19 | \( 1 - 4.39T + 19T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 + 5.32T + 29T^{2} \) |
| 31 | \( 1 - 2.06T + 31T^{2} \) |
| 37 | \( 1 - 2.29T + 37T^{2} \) |
| 41 | \( 1 - 3.69T + 41T^{2} \) |
| 43 | \( 1 - 6.38T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 - 7.84T + 53T^{2} \) |
| 59 | \( 1 + 8.58T + 59T^{2} \) |
| 61 | \( 1 - 3.08T + 61T^{2} \) |
| 67 | \( 1 + 5.38T + 67T^{2} \) |
| 71 | \( 1 + 0.409T + 71T^{2} \) |
| 73 | \( 1 + 9.61T + 73T^{2} \) |
| 79 | \( 1 - 4.71T + 79T^{2} \) |
| 83 | \( 1 - 7.35T + 83T^{2} \) |
| 89 | \( 1 + 0.833T + 89T^{2} \) |
| 97 | \( 1 - 2.28T + 97T^{2} \) |
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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
−7.79252413977028355085980939487, −7.36963201837193913801703510036, −6.24145510914310110364279463549, −5.75046346151910122595692860259, −5.24575279441959426604080618234, −4.16401319223165457071091434324, −3.61568315390265502236665449013, −2.68093447380220594201388815737, −1.47404208304694641534102340694, −0.810679312653126279019806097586,
0.810679312653126279019806097586, 1.47404208304694641534102340694, 2.68093447380220594201388815737, 3.61568315390265502236665449013, 4.16401319223165457071091434324, 5.24575279441959426604080618234, 5.75046346151910122595692860259, 6.24145510914310110364279463549, 7.36963201837193913801703510036, 7.79252413977028355085980939487
