| L(s) = 1 | − 1.33·2-s − 0.210·4-s + 0.660·5-s + 7-s + 2.95·8-s − 0.883·10-s + 2.50·11-s + 3.59·13-s − 1.33·14-s − 3.53·16-s − 0.647·17-s + 4.90·19-s − 0.139·20-s − 3.34·22-s − 4.27·23-s − 4.56·25-s − 4.81·26-s − 0.210·28-s + 7.60·29-s − 2.73·31-s − 1.18·32-s + 0.866·34-s + 0.660·35-s + 0.693·37-s − 6.55·38-s + 1.95·40-s + 11.1·41-s + ⋯ |
| L(s) = 1 | − 0.945·2-s − 0.105·4-s + 0.295·5-s + 0.377·7-s + 1.04·8-s − 0.279·10-s + 0.754·11-s + 0.997·13-s − 0.357·14-s − 0.883·16-s − 0.157·17-s + 1.12·19-s − 0.0311·20-s − 0.713·22-s − 0.892·23-s − 0.912·25-s − 0.943·26-s − 0.0398·28-s + 1.41·29-s − 0.491·31-s − 0.209·32-s + 0.148·34-s + 0.111·35-s + 0.114·37-s − 1.06·38-s + 0.308·40-s + 1.74·41-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
| \(L(1)\) |
\(\approx\) |
\(1.469874113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.469874113\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.33T + 2T^{2} \) |
| 5 | \( 1 - 0.660T + 5T^{2} \) |
| 11 | \( 1 - 2.50T + 11T^{2} \) |
| 13 | \( 1 - 3.59T + 13T^{2} \) |
| 17 | \( 1 + 0.647T + 17T^{2} \) |
| 19 | \( 1 - 4.90T + 19T^{2} \) |
| 23 | \( 1 + 4.27T + 23T^{2} \) |
| 29 | \( 1 - 7.60T + 29T^{2} \) |
| 31 | \( 1 + 2.73T + 31T^{2} \) |
| 37 | \( 1 - 0.693T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + 1.57T + 43T^{2} \) |
| 47 | \( 1 - 6.59T + 47T^{2} \) |
| 53 | \( 1 + 7.27T + 53T^{2} \) |
| 59 | \( 1 - 9.88T + 59T^{2} \) |
| 61 | \( 1 - 0.382T + 61T^{2} \) |
| 67 | \( 1 + 4.37T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 - 4.43T + 73T^{2} \) |
| 79 | \( 1 + 6.44T + 79T^{2} \) |
| 83 | \( 1 - 4.82T + 83T^{2} \) |
| 89 | \( 1 - 9.63T + 89T^{2} \) |
| 97 | \( 1 + 0.182T + 97T^{2} \) |
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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
−7.977452955902796860619264098330, −7.39586384208226808665513826167, −6.51361215726355859577773902263, −5.87026845681061961782315213762, −5.06273757741818102653092849425, −4.18707327403329687870219549011, −3.65748017128087543311595755010, −2.38018814326814476900044126691, −1.45946134405110426953703795188, −0.78058242816577649985403663605,
0.78058242816577649985403663605, 1.45946134405110426953703795188, 2.38018814326814476900044126691, 3.65748017128087543311595755010, 4.18707327403329687870219549011, 5.06273757741818102653092849425, 5.87026845681061961782315213762, 6.51361215726355859577773902263, 7.39586384208226808665513826167, 7.977452955902796860619264098330
