| L(s) = 1 | − 0.414·2-s + 1.41·3-s − 1.82·4-s + 1.82·5-s − 0.585·6-s + 1.58·8-s − 0.999·9-s − 0.757·10-s − 0.585·11-s − 2.58·12-s + 2.58·15-s + 3·16-s − 5.82·17-s + 0.414·18-s − 6·19-s − 3.34·20-s + 0.242·22-s − 1.41·23-s + 2.24·24-s − 1.65·25-s − 5.65·27-s + 4.17·29-s − 1.07·30-s − 2.58·31-s − 4.41·32-s − 0.828·33-s + 2.41·34-s + ⋯ |
| L(s) = 1 | − 0.292·2-s + 0.816·3-s − 0.914·4-s + 0.817·5-s − 0.239·6-s + 0.560·8-s − 0.333·9-s − 0.239·10-s − 0.176·11-s − 0.746·12-s + 0.667·15-s + 0.750·16-s − 1.41·17-s + 0.0976·18-s − 1.37·19-s − 0.747·20-s + 0.0517·22-s − 0.294·23-s + 0.457·24-s − 0.331·25-s − 1.08·27-s + 0.774·29-s − 0.195·30-s − 0.464·31-s − 0.780·32-s − 0.144·33-s + 0.414·34-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
| \(L(1)\) |
\(\approx\) |
\(1.550900082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.550900082\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.414T + 2T^{2} \) |
| 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 - 1.82T + 5T^{2} \) |
| 11 | \( 1 + 0.585T + 11T^{2} \) |
| 17 | \( 1 + 5.82T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 - 4.17T + 29T^{2} \) |
| 31 | \( 1 + 2.58T + 31T^{2} \) |
| 37 | \( 1 - 9.48T + 37T^{2} \) |
| 41 | \( 1 + 0.171T + 41T^{2} \) |
| 43 | \( 1 - 3.41T + 43T^{2} \) |
| 47 | \( 1 - 3.65T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 4.17T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 1.75T + 79T^{2} \) |
| 83 | \( 1 + 1.07T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.062222353659990587775156118440, −7.31110255312351827616791396550, −6.29068717208586185065002546046, −5.85922356558574125060058220885, −4.87692735864111318652099031482, −4.26775875512194766579236434408, −3.53137087809039598092457285390, −2.37155732724389579763550158198, −2.05615359259326968854148660441, −0.59430514561640484423282045045,
0.59430514561640484423282045045, 2.05615359259326968854148660441, 2.37155732724389579763550158198, 3.53137087809039598092457285390, 4.26775875512194766579236434408, 4.87692735864111318652099031482, 5.85922356558574125060058220885, 6.29068717208586185065002546046, 7.31110255312351827616791396550, 8.062222353659990587775156118440
