| L(s) = 1 | + 2.45·2-s + 3-s + 4.04·4-s − 3.55·5-s + 2.45·6-s − 7-s + 5.03·8-s + 9-s − 8.74·10-s + 0.397·11-s + 4.04·12-s + 2.00·13-s − 2.45·14-s − 3.55·15-s + 4.29·16-s − 5.34·17-s + 2.45·18-s − 1.72·19-s − 14.3·20-s − 21-s + 0.978·22-s − 1.19·23-s + 5.03·24-s + 7.63·25-s + 4.92·26-s + 27-s − 4.04·28-s + ⋯ |
| L(s) = 1 | + 1.73·2-s + 0.577·3-s + 2.02·4-s − 1.58·5-s + 1.00·6-s − 0.377·7-s + 1.78·8-s + 0.333·9-s − 2.76·10-s + 0.119·11-s + 1.16·12-s + 0.555·13-s − 0.657·14-s − 0.917·15-s + 1.07·16-s − 1.29·17-s + 0.579·18-s − 0.394·19-s − 3.21·20-s − 0.218·21-s + 0.208·22-s − 0.248·23-s + 1.02·24-s + 1.52·25-s + 0.966·26-s + 0.192·27-s − 0.765·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 2.45T + 2T^{2} \) |
| 5 | \( 1 + 3.55T + 5T^{2} \) |
| 11 | \( 1 - 0.397T + 11T^{2} \) |
| 13 | \( 1 - 2.00T + 13T^{2} \) |
| 17 | \( 1 + 5.34T + 17T^{2} \) |
| 19 | \( 1 + 1.72T + 19T^{2} \) |
| 23 | \( 1 + 1.19T + 23T^{2} \) |
| 29 | \( 1 + 5.34T + 29T^{2} \) |
| 31 | \( 1 - 7.87T + 31T^{2} \) |
| 37 | \( 1 + 0.507T + 37T^{2} \) |
| 41 | \( 1 + 4.86T + 41T^{2} \) |
| 43 | \( 1 + 6.36T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 6.18T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 7.67T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 8.56T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 - 7.75T + 83T^{2} \) |
| 89 | \( 1 + 18.3T + 89T^{2} \) |
| 97 | \( 1 - 4.22T + 97T^{2} \) |
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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
−7.26212032022906511850048010994, −6.64700017106198635564376533772, −6.19846701192538535221481376411, −5.04969713503766896239508264012, −4.44714673498747352189401688150, −3.93081807966064773554434268812, −3.37686627911390693862211979352, −2.73520004577869733183033219382, −1.70212835305803305600006963760, 0,
1.70212835305803305600006963760, 2.73520004577869733183033219382, 3.37686627911390693862211979352, 3.93081807966064773554434268812, 4.44714673498747352189401688150, 5.04969713503766896239508264012, 6.19846701192538535221481376411, 6.64700017106198635564376533772, 7.26212032022906511850048010994
