| L(s) = 1 | − 1.72·2-s + 0.962·4-s − 2.01·5-s + 7-s + 1.78·8-s + 3.46·10-s − 3.24·11-s − 2.61·13-s − 1.72·14-s − 4.99·16-s + 1.03·17-s − 7.18·19-s − 1.94·20-s + 5.59·22-s + 8.35·23-s − 0.939·25-s + 4.50·26-s + 0.962·28-s + 2.77·29-s + 7.77·31-s + 5.03·32-s − 1.78·34-s − 2.01·35-s − 0.894·37-s + 12.3·38-s − 3.59·40-s − 9.16·41-s + ⋯ |
| L(s) = 1 | − 1.21·2-s + 0.481·4-s − 0.901·5-s + 0.377·7-s + 0.631·8-s + 1.09·10-s − 0.979·11-s − 0.726·13-s − 0.460·14-s − 1.24·16-s + 0.251·17-s − 1.64·19-s − 0.433·20-s + 1.19·22-s + 1.74·23-s − 0.187·25-s + 0.883·26-s + 0.181·28-s + 0.514·29-s + 1.39·31-s + 0.889·32-s − 0.305·34-s − 0.340·35-s − 0.147·37-s + 2.00·38-s − 0.568·40-s − 1.43·41-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 | \( 1 + 1.72T + 2T^{2} \) |
| 5 | \( 1 + 2.01T + 5T^{2} \) |
| 11 | \( 1 + 3.24T + 11T^{2} \) |
| 13 | \( 1 + 2.61T + 13T^{2} \) |
| 17 | \( 1 - 1.03T + 17T^{2} \) |
| 19 | \( 1 + 7.18T + 19T^{2} \) |
| 23 | \( 1 - 8.35T + 23T^{2} \) |
| 29 | \( 1 - 2.77T + 29T^{2} \) |
| 31 | \( 1 - 7.77T + 31T^{2} \) |
| 37 | \( 1 + 0.894T + 37T^{2} \) |
| 41 | \( 1 + 9.16T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 1.21T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 1.59T + 61T^{2} \) |
| 67 | \( 1 + 6.57T + 67T^{2} \) |
| 71 | \( 1 + 5.66T + 71T^{2} \) |
| 73 | \( 1 - 16.1T + 73T^{2} \) |
| 79 | \( 1 - 6.54T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 - 2.20T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−7.76827343748548971096968467857, −7.05512843952259616579383739280, −6.48103933030031695911610147931, −5.15575972689832467225071216073, −4.75380732346349234375486358721, −3.97677932473269508205926506314, −2.85308345544661768940638754823, −2.09167875246182332999415980188, −0.896268805103735423933126610693, 0,
0.896268805103735423933126610693, 2.09167875246182332999415980188, 2.85308345544661768940638754823, 3.97677932473269508205926506314, 4.75380732346349234375486358721, 5.15575972689832467225071216073, 6.48103933030031695911610147931, 7.05512843952259616579383739280, 7.76827343748548971096968467857
