| L(s) = 1 | + 1.73·2-s + 0.732·3-s + 0.999·4-s − 1.73·5-s + 1.26·6-s − 1.73·8-s − 2.46·9-s − 2.99·10-s − 4.73·11-s + 0.732·12-s − 1.26·15-s − 5·16-s + 4.26·17-s − 4.26·18-s + 2·19-s − 1.73·20-s − 8.19·22-s + 1.26·23-s − 1.26·24-s − 2.00·25-s − 4·27-s − 3·29-s − 2.19·30-s − 6.19·31-s − 5.19·32-s − 3.46·33-s + 7.39·34-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.422·3-s + 0.499·4-s − 0.774·5-s + 0.517·6-s − 0.612·8-s − 0.821·9-s − 0.948·10-s − 1.42·11-s + 0.211·12-s − 0.327·15-s − 1.25·16-s + 1.03·17-s − 1.00·18-s + 0.458·19-s − 0.387·20-s − 1.74·22-s + 0.264·23-s − 0.258·24-s − 0.400·25-s − 0.769·27-s − 0.557·29-s − 0.400·30-s − 1.11·31-s − 0.918·32-s − 0.603·33-s + 1.26·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.257455115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.257455115\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 3 | \( 1 - 0.732T + 3T^{2} \) |
| 5 | \( 1 + 1.73T + 5T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 17 | \( 1 - 4.26T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 1.26T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 6.19T + 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 - 5.19T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 + 0.928T + 47T^{2} \) |
| 53 | \( 1 - 3.92T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 + 4.19T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 7.19T + 73T^{2} \) |
| 79 | \( 1 - 5.80T + 79T^{2} \) |
| 83 | \( 1 - 8.19T + 83T^{2} \) |
| 89 | \( 1 + 0.928T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−7.74278921476140862347399772201, −7.25434189455838319139100948948, −6.09356677870107854196906219099, −5.45598887328111590311392728950, −5.19549790539972441915303791422, −4.05246919670495923940146453428, −3.66260283536706829563095684577, −2.82987671572009940831867860886, −2.34949158270939851314979319349, −0.56409947913096698444824593142,
0.56409947913096698444824593142, 2.34949158270939851314979319349, 2.82987671572009940831867860886, 3.66260283536706829563095684577, 4.05246919670495923940146453428, 5.19549790539972441915303791422, 5.45598887328111590311392728950, 6.09356677870107854196906219099, 7.25434189455838319139100948948, 7.74278921476140862347399772201
