| L(s) = 1 | + 1.94·2-s + 1.77·4-s − 3.66·5-s + 7-s − 0.445·8-s − 7.11·10-s − 2.76·11-s − 3.90·13-s + 1.94·14-s − 4.40·16-s − 3.65·17-s − 2.93·19-s − 6.49·20-s − 5.36·22-s + 5.90·23-s + 8.44·25-s − 7.57·26-s + 1.77·28-s − 1.00·29-s − 0.885·31-s − 7.66·32-s − 7.09·34-s − 3.66·35-s + 5.97·37-s − 5.70·38-s + 1.63·40-s − 7.04·41-s + ⋯ |
| L(s) = 1 | + 1.37·2-s + 0.885·4-s − 1.63·5-s + 0.377·7-s − 0.157·8-s − 2.25·10-s − 0.833·11-s − 1.08·13-s + 0.518·14-s − 1.10·16-s − 0.886·17-s − 0.673·19-s − 1.45·20-s − 1.14·22-s + 1.23·23-s + 1.68·25-s − 1.48·26-s + 0.334·28-s − 0.187·29-s − 0.158·31-s − 1.35·32-s − 1.21·34-s − 0.619·35-s + 0.981·37-s − 0.924·38-s + 0.258·40-s − 1.09·41-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
| \(L(1)\) |
\(\approx\) |
\(1.640607611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.640607611\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.94T + 2T^{2} \) |
| 5 | \( 1 + 3.66T + 5T^{2} \) |
| 11 | \( 1 + 2.76T + 11T^{2} \) |
| 13 | \( 1 + 3.90T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 + 2.93T + 19T^{2} \) |
| 23 | \( 1 - 5.90T + 23T^{2} \) |
| 29 | \( 1 + 1.00T + 29T^{2} \) |
| 31 | \( 1 + 0.885T + 31T^{2} \) |
| 37 | \( 1 - 5.97T + 37T^{2} \) |
| 41 | \( 1 + 7.04T + 41T^{2} \) |
| 43 | \( 1 - 5.95T + 43T^{2} \) |
| 47 | \( 1 - 5.51T + 47T^{2} \) |
| 53 | \( 1 + 4.27T + 53T^{2} \) |
| 59 | \( 1 - 7.36T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 8.14T + 67T^{2} \) |
| 71 | \( 1 + 4.06T + 71T^{2} \) |
| 73 | \( 1 - 1.01T + 73T^{2} \) |
| 79 | \( 1 - 1.03T + 79T^{2} \) |
| 83 | \( 1 + 3.97T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−7.57415099767645425530090715996, −7.14657056841316009861308689329, −6.45061625141016088231972633509, −5.36855464243524206424478467446, −4.89682905328664486872609027469, −4.32342296283999083550607875084, −3.77022871821564165801963028420, −2.86447757080285801276647311951, −2.29360290234172728368705552752, −0.48140514721191797816145867654,
0.48140514721191797816145867654, 2.29360290234172728368705552752, 2.86447757080285801276647311951, 3.77022871821564165801963028420, 4.32342296283999083550607875084, 4.89682905328664486872609027469, 5.36855464243524206424478467446, 6.45061625141016088231972633509, 7.14657056841316009861308689329, 7.57415099767645425530090715996
