| L(s) = 1 | − 2.58·2-s + 1.51·3-s + 4.68·4-s − 3.91·5-s − 3.92·6-s − 1.90·7-s − 6.94·8-s − 0.699·9-s + 10.1·10-s + 3.75·11-s + 7.10·12-s + 0.142·13-s + 4.93·14-s − 5.94·15-s + 8.58·16-s + 0.966·17-s + 1.80·18-s + 3.91·19-s − 18.3·20-s − 2.89·21-s − 9.71·22-s + 8.94·23-s − 10.5·24-s + 10.3·25-s − 0.368·26-s − 5.61·27-s − 8.93·28-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 0.875·3-s + 2.34·4-s − 1.75·5-s − 1.60·6-s − 0.720·7-s − 2.45·8-s − 0.233·9-s + 3.20·10-s + 1.13·11-s + 2.05·12-s + 0.0395·13-s + 1.31·14-s − 1.53·15-s + 2.14·16-s + 0.234·17-s + 0.426·18-s + 0.897·19-s − 4.10·20-s − 0.631·21-s − 2.07·22-s + 1.86·23-s − 2.15·24-s + 2.06·25-s − 0.0723·26-s − 1.07·27-s − 1.68·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6671722745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6671722745\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 53 | \( 1 + T \) |
| 151 | \( 1 - T \) |
| good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 3 | \( 1 - 1.51T + 3T^{2} \) |
| 5 | \( 1 + 3.91T + 5T^{2} \) |
| 7 | \( 1 + 1.90T + 7T^{2} \) |
| 11 | \( 1 - 3.75T + 11T^{2} \) |
| 13 | \( 1 - 0.142T + 13T^{2} \) |
| 17 | \( 1 - 0.966T + 17T^{2} \) |
| 19 | \( 1 - 3.91T + 19T^{2} \) |
| 23 | \( 1 - 8.94T + 23T^{2} \) |
| 29 | \( 1 + 1.33T + 29T^{2} \) |
| 31 | \( 1 - 5.78T + 31T^{2} \) |
| 37 | \( 1 - 8.39T + 37T^{2} \) |
| 41 | \( 1 + 7.06T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 5.88T + 47T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 4.88T + 67T^{2} \) |
| 71 | \( 1 + 5.62T + 71T^{2} \) |
| 73 | \( 1 - 2.67T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + 4.48T + 83T^{2} \) |
| 89 | \( 1 + 4.37T + 89T^{2} \) |
| 97 | \( 1 - 7.22T + 97T^{2} \) |
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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
−8.079232039695212068254118218390, −7.34586045275966860949877288580, −6.93733377542909957507617241948, −6.31778506773773578026875764938, −4.99201582909629591383158576649, −3.81078390717186082288728721870, −3.22782969500366565350270762877, −2.75610432884035328040383401344, −1.36772164421440277389621194783, −0.55408230845751597578255457093,
0.55408230845751597578255457093, 1.36772164421440277389621194783, 2.75610432884035328040383401344, 3.22782969500366565350270762877, 3.81078390717186082288728721870, 4.99201582909629591383158576649, 6.31778506773773578026875764938, 6.93733377542909957507617241948, 7.34586045275966860949877288580, 8.079232039695212068254118218390
