| L(s) = 1 | + 2.66·2-s + 5.08·4-s + 1.08·5-s + 7-s + 8.22·8-s + 2.89·10-s + 4.06·11-s − 1.24·13-s + 2.66·14-s + 11.7·16-s + 1.93·17-s − 4.54·19-s + 5.52·20-s + 10.8·22-s + 4.23·23-s − 3.82·25-s − 3.31·26-s + 5.08·28-s + 10.6·29-s − 7.36·31-s + 14.7·32-s + 5.16·34-s + 1.08·35-s − 7.14·37-s − 12.0·38-s + 8.93·40-s + 1.34·41-s + ⋯ |
| L(s) = 1 | + 1.88·2-s + 2.54·4-s + 0.485·5-s + 0.377·7-s + 2.90·8-s + 0.913·10-s + 1.22·11-s − 0.344·13-s + 0.711·14-s + 2.93·16-s + 0.470·17-s − 1.04·19-s + 1.23·20-s + 2.30·22-s + 0.882·23-s − 0.764·25-s − 0.649·26-s + 0.961·28-s + 1.97·29-s − 1.32·31-s + 2.61·32-s + 0.885·34-s + 0.183·35-s − 1.17·37-s − 1.96·38-s + 1.41·40-s + 0.210·41-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
| \(L(1)\) |
\(\approx\) |
\(9.450742231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.450742231\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.66T + 2T^{2} \) |
| 5 | \( 1 - 1.08T + 5T^{2} \) |
| 11 | \( 1 - 4.06T + 11T^{2} \) |
| 13 | \( 1 + 1.24T + 13T^{2} \) |
| 17 | \( 1 - 1.93T + 17T^{2} \) |
| 19 | \( 1 + 4.54T + 19T^{2} \) |
| 23 | \( 1 - 4.23T + 23T^{2} \) |
| 29 | \( 1 - 10.6T + 29T^{2} \) |
| 31 | \( 1 + 7.36T + 31T^{2} \) |
| 37 | \( 1 + 7.14T + 37T^{2} \) |
| 41 | \( 1 - 1.34T + 41T^{2} \) |
| 43 | \( 1 - 0.657T + 43T^{2} \) |
| 47 | \( 1 + 1.90T + 47T^{2} \) |
| 53 | \( 1 - 7.63T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 5.42T + 61T^{2} \) |
| 67 | \( 1 - 3.79T + 67T^{2} \) |
| 71 | \( 1 + 5.32T + 71T^{2} \) |
| 73 | \( 1 - 7.99T + 73T^{2} \) |
| 79 | \( 1 + 2.53T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + 5.06T + 97T^{2} \) |
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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
−7.45286364539658012288759426990, −6.72297857650143299581443431061, −6.42908106264410259018090487765, −5.51008166469372912545265357335, −5.10610671336853416756319450528, −4.21843885112366374692849433194, −3.78418216198796995811016529930, −2.84499996712138392494058702313, −2.07358903611050027109459451475, −1.29054061159942801040330907227,
1.29054061159942801040330907227, 2.07358903611050027109459451475, 2.84499996712138392494058702313, 3.78418216198796995811016529930, 4.21843885112366374692849433194, 5.10610671336853416756319450528, 5.51008166469372912545265357335, 6.42908106264410259018090487765, 6.72297857650143299581443431061, 7.45286364539658012288759426990
