| L(s) = 1 | − 2.62·2-s − 3-s + 4.88·4-s − 1.77·5-s + 2.62·6-s − 7-s − 7.57·8-s + 9-s + 4.66·10-s − 0.992·11-s − 4.88·12-s + 0.501·13-s + 2.62·14-s + 1.77·15-s + 10.1·16-s − 4.64·17-s − 2.62·18-s + 0.340·19-s − 8.68·20-s + 21-s + 2.60·22-s + 1.71·23-s + 7.57·24-s − 1.84·25-s − 1.31·26-s − 27-s − 4.88·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 0.577·3-s + 2.44·4-s − 0.794·5-s + 1.07·6-s − 0.377·7-s − 2.67·8-s + 0.333·9-s + 1.47·10-s − 0.299·11-s − 1.41·12-s + 0.139·13-s + 0.701·14-s + 0.458·15-s + 2.52·16-s − 1.12·17-s − 0.618·18-s + 0.0781·19-s − 1.94·20-s + 0.218·21-s + 0.555·22-s + 0.357·23-s + 1.54·24-s − 0.368·25-s − 0.258·26-s − 0.192·27-s − 0.923·28-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
| \(L(1)\) |
\(\approx\) |
\(0.03901274255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03901274255\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.62T + 2T^{2} \) |
| 5 | \( 1 + 1.77T + 5T^{2} \) |
| 11 | \( 1 + 0.992T + 11T^{2} \) |
| 13 | \( 1 - 0.501T + 13T^{2} \) |
| 17 | \( 1 + 4.64T + 17T^{2} \) |
| 19 | \( 1 - 0.340T + 19T^{2} \) |
| 23 | \( 1 - 1.71T + 23T^{2} \) |
| 29 | \( 1 + 9.58T + 29T^{2} \) |
| 31 | \( 1 + 9.30T + 31T^{2} \) |
| 37 | \( 1 + 3.76T + 37T^{2} \) |
| 41 | \( 1 - 0.0486T + 41T^{2} \) |
| 43 | \( 1 - 1.77T + 43T^{2} \) |
| 47 | \( 1 + 5.26T + 47T^{2} \) |
| 53 | \( 1 - 1.88T + 53T^{2} \) |
| 59 | \( 1 + 7.26T + 59T^{2} \) |
| 61 | \( 1 + 1.78T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 + 8.17T + 71T^{2} \) |
| 73 | \( 1 - 6.13T + 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 5.45T + 89T^{2} \) |
| 97 | \( 1 - 7.64T + 97T^{2} \) |
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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
−7.83728078476321689278972577794, −7.25955047991834472317768717891, −6.85759120968556704743544110168, −6.02413711623336801020034137680, −5.31718657734243152611404544994, −4.12244917659035187648140846091, −3.34322804340204778590478644953, −2.26258515960120914423274490107, −1.47396102221927753808961758578, −0.13443982928028032419833525642,
0.13443982928028032419833525642, 1.47396102221927753808961758578, 2.26258515960120914423274490107, 3.34322804340204778590478644953, 4.12244917659035187648140846091, 5.31718657734243152611404544994, 6.02413711623336801020034137680, 6.85759120968556704743544110168, 7.25955047991834472317768717891, 7.83728078476321689278972577794
