| L(s) = 1 | − 0.487·2-s − 1.76·4-s + 0.708·5-s − 7-s + 1.83·8-s − 0.345·10-s + 4.44·11-s − 4.01·13-s + 0.487·14-s + 2.63·16-s + 1.16·17-s + 0.231·19-s − 1.24·20-s − 2.16·22-s + 6.97·23-s − 4.49·25-s + 1.95·26-s + 1.76·28-s + 0.276·29-s − 6.86·31-s − 4.94·32-s − 0.568·34-s − 0.708·35-s − 3.38·37-s − 0.112·38-s + 1.29·40-s + 7.55·41-s + ⋯ |
| L(s) = 1 | − 0.344·2-s − 0.881·4-s + 0.316·5-s − 0.377·7-s + 0.648·8-s − 0.109·10-s + 1.34·11-s − 1.11·13-s + 0.130·14-s + 0.657·16-s + 0.282·17-s + 0.0530·19-s − 0.279·20-s − 0.461·22-s + 1.45·23-s − 0.899·25-s + 0.384·26-s + 0.333·28-s + 0.0513·29-s − 1.23·31-s − 0.874·32-s − 0.0974·34-s − 0.119·35-s − 0.556·37-s − 0.0182·38-s + 0.205·40-s + 1.18·41-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
| \(L(1)\) |
\(\approx\) |
\(1.292989086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.292989086\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.487T + 2T^{2} \) |
| 5 | \( 1 - 0.708T + 5T^{2} \) |
| 11 | \( 1 - 4.44T + 11T^{2} \) |
| 13 | \( 1 + 4.01T + 13T^{2} \) |
| 17 | \( 1 - 1.16T + 17T^{2} \) |
| 19 | \( 1 - 0.231T + 19T^{2} \) |
| 23 | \( 1 - 6.97T + 23T^{2} \) |
| 29 | \( 1 - 0.276T + 29T^{2} \) |
| 31 | \( 1 + 6.86T + 31T^{2} \) |
| 37 | \( 1 + 3.38T + 37T^{2} \) |
| 41 | \( 1 - 7.55T + 41T^{2} \) |
| 43 | \( 1 - 5.27T + 43T^{2} \) |
| 47 | \( 1 + 5.93T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 - 9.69T + 59T^{2} \) |
| 61 | \( 1 - 9.65T + 61T^{2} \) |
| 67 | \( 1 + 5.37T + 67T^{2} \) |
| 71 | \( 1 - 2.44T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 9.67T + 79T^{2} \) |
| 83 | \( 1 + 4.65T + 83T^{2} \) |
| 89 | \( 1 - 2.71T + 89T^{2} \) |
| 97 | \( 1 - 5.35T + 97T^{2} \) |
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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
−7.78178438459737303901548552174, −7.21899371803753489579947417562, −6.58181480026252760281085776056, −5.58944074518011382164487281767, −5.15102942254853031483917388932, −4.16480618740183966216734487973, −3.70515629209910908278434385862, −2.64164025963936273404248505252, −1.57478995665219802415584106908, −0.62769443423472969271367026353,
0.62769443423472969271367026353, 1.57478995665219802415584106908, 2.64164025963936273404248505252, 3.70515629209910908278434385862, 4.16480618740183966216734487973, 5.15102942254853031483917388932, 5.58944074518011382164487281767, 6.58181480026252760281085776056, 7.21899371803753489579947417562, 7.78178438459737303901548552174
