| L(s) = 1 | − 2.62·2-s − 2.48·3-s + 4.88·4-s − 3.25·5-s + 6.51·6-s + 0.610·7-s − 7.57·8-s + 3.16·9-s + 8.55·10-s + 6.56·11-s − 12.1·12-s − 0.553·13-s − 1.60·14-s + 8.08·15-s + 10.0·16-s + 6.81·17-s − 8.29·18-s + 7.03·19-s − 15.9·20-s − 1.51·21-s − 17.2·22-s − 1.85·23-s + 18.7·24-s + 5.61·25-s + 1.45·26-s − 0.399·27-s + 2.98·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 1.43·3-s + 2.44·4-s − 1.45·5-s + 2.65·6-s + 0.230·7-s − 2.67·8-s + 1.05·9-s + 2.70·10-s + 1.98·11-s − 3.50·12-s − 0.153·13-s − 0.428·14-s + 2.08·15-s + 2.52·16-s + 1.65·17-s − 1.95·18-s + 1.61·19-s − 3.56·20-s − 0.330·21-s − 3.67·22-s − 0.386·23-s + 3.83·24-s + 1.12·25-s + 0.285·26-s − 0.0769·27-s + 0.564·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 53 | \( 1 + T \) |
| 151 | \( 1 + T \) |
| good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 3 | \( 1 + 2.48T + 3T^{2} \) |
| 5 | \( 1 + 3.25T + 5T^{2} \) |
| 7 | \( 1 - 0.610T + 7T^{2} \) |
| 11 | \( 1 - 6.56T + 11T^{2} \) |
| 13 | \( 1 + 0.553T + 13T^{2} \) |
| 17 | \( 1 - 6.81T + 17T^{2} \) |
| 19 | \( 1 - 7.03T + 19T^{2} \) |
| 23 | \( 1 + 1.85T + 23T^{2} \) |
| 29 | \( 1 - 6.42T + 29T^{2} \) |
| 31 | \( 1 + 1.64T + 31T^{2} \) |
| 37 | \( 1 + 2.92T + 37T^{2} \) |
| 41 | \( 1 - 4.41T + 41T^{2} \) |
| 43 | \( 1 + 2.75T + 43T^{2} \) |
| 47 | \( 1 - 0.122T + 47T^{2} \) |
| 59 | \( 1 - 2.05T + 59T^{2} \) |
| 61 | \( 1 + 8.14T + 61T^{2} \) |
| 67 | \( 1 + 5.00T + 67T^{2} \) |
| 71 | \( 1 - 0.658T + 71T^{2} \) |
| 73 | \( 1 - 5.08T + 73T^{2} \) |
| 79 | \( 1 + 9.53T + 79T^{2} \) |
| 83 | \( 1 + 0.567T + 83T^{2} \) |
| 89 | \( 1 + 2.38T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−7.53016042808986160194863006084, −7.02969708163144751998470528674, −6.40774446369273254992552318233, −5.71379078738843455993072542302, −4.77288285338310797229380029975, −3.78000582173280222247936367228, −3.06057803897415047439880872940, −1.26979042821043700383178961490, −1.10591762912960014767883601668, 0,
1.10591762912960014767883601668, 1.26979042821043700383178961490, 3.06057803897415047439880872940, 3.78000582173280222247936367228, 4.77288285338310797229380029975, 5.71379078738843455993072542302, 6.40774446369273254992552318233, 7.02969708163144751998470528674, 7.53016042808986160194863006084
