| L(s) = 1 | + 0.823·2-s − 1.32·4-s + 3.74·5-s − 7-s − 2.73·8-s + 3.07·10-s − 5.51·11-s + 3.05·13-s − 0.823·14-s + 0.394·16-s + 2.99·17-s − 8.18·19-s − 4.94·20-s − 4.53·22-s + 8.49·23-s + 8.99·25-s + 2.51·26-s + 1.32·28-s − 2.18·29-s − 10.0·31-s + 5.79·32-s + 2.46·34-s − 3.74·35-s + 8.15·37-s − 6.73·38-s − 10.2·40-s − 5.46·41-s + ⋯ |
| L(s) = 1 | + 0.581·2-s − 0.661·4-s + 1.67·5-s − 0.377·7-s − 0.966·8-s + 0.973·10-s − 1.66·11-s + 0.846·13-s − 0.219·14-s + 0.0986·16-s + 0.726·17-s − 1.87·19-s − 1.10·20-s − 0.966·22-s + 1.77·23-s + 1.79·25-s + 0.492·26-s + 0.249·28-s − 0.406·29-s − 1.80·31-s + 1.02·32-s + 0.422·34-s − 0.632·35-s + 1.33·37-s − 1.09·38-s − 1.61·40-s − 0.853·41-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 0.823T + 2T^{2} \) |
| 5 | \( 1 - 3.74T + 5T^{2} \) |
| 11 | \( 1 + 5.51T + 11T^{2} \) |
| 13 | \( 1 - 3.05T + 13T^{2} \) |
| 17 | \( 1 - 2.99T + 17T^{2} \) |
| 19 | \( 1 + 8.18T + 19T^{2} \) |
| 23 | \( 1 - 8.49T + 23T^{2} \) |
| 29 | \( 1 + 2.18T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 8.15T + 37T^{2} \) |
| 41 | \( 1 + 5.46T + 41T^{2} \) |
| 43 | \( 1 + 4.04T + 43T^{2} \) |
| 47 | \( 1 + 3.81T + 47T^{2} \) |
| 53 | \( 1 - 5.00T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 + 8.19T + 61T^{2} \) |
| 67 | \( 1 + 5.69T + 67T^{2} \) |
| 71 | \( 1 + 7.02T + 71T^{2} \) |
| 73 | \( 1 + 2.25T + 73T^{2} \) |
| 79 | \( 1 + 1.56T + 79T^{2} \) |
| 83 | \( 1 + 5.39T + 83T^{2} \) |
| 89 | \( 1 + 2.98T + 89T^{2} \) |
| 97 | \( 1 + 7.99T + 97T^{2} \) |
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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
−7.37370157289543967425192180326, −6.50358733229036768497737162480, −5.89214867528181647443222619720, −5.39907458610745568784509230191, −4.93668720821367733464521978303, −3.93047790151904993453558552351, −3.02487057619582339713580828377, −2.44379431195505825946881107473, −1.38951589055103835488383586935, 0,
1.38951589055103835488383586935, 2.44379431195505825946881107473, 3.02487057619582339713580828377, 3.93047790151904993453558552351, 4.93668720821367733464521978303, 5.39907458610745568784509230191, 5.89214867528181647443222619720, 6.50358733229036768497737162480, 7.37370157289543967425192180326
