| L(s) = 1 | + 2.68·2-s + 1.84·3-s + 5.22·4-s − 2.39·5-s + 4.95·6-s + 0.545·7-s + 8.66·8-s + 0.402·9-s − 6.43·10-s + 1.86·11-s + 9.63·12-s + 13-s + 1.46·14-s − 4.42·15-s + 12.8·16-s + 4.21·17-s + 1.08·18-s − 2.74·19-s − 12.5·20-s + 1.00·21-s + 5.00·22-s − 6.86·23-s + 15.9·24-s + 0.742·25-s + 2.68·26-s − 4.79·27-s + 2.85·28-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 1.06·3-s + 2.61·4-s − 1.07·5-s + 2.02·6-s + 0.206·7-s + 3.06·8-s + 0.134·9-s − 2.03·10-s + 0.561·11-s + 2.78·12-s + 0.277·13-s + 0.392·14-s − 1.14·15-s + 3.20·16-s + 1.02·17-s + 0.255·18-s − 0.629·19-s − 2.79·20-s + 0.219·21-s + 1.06·22-s − 1.43·23-s + 3.26·24-s + 0.148·25-s + 0.527·26-s − 0.922·27-s + 0.538·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.560808863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.560808863\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 2.68T + 2T^{2} \) |
| 3 | \( 1 - 1.84T + 3T^{2} \) |
| 5 | \( 1 + 2.39T + 5T^{2} \) |
| 7 | \( 1 - 0.545T + 7T^{2} \) |
| 11 | \( 1 - 1.86T + 11T^{2} \) |
| 17 | \( 1 - 4.21T + 17T^{2} \) |
| 19 | \( 1 + 2.74T + 19T^{2} \) |
| 23 | \( 1 + 6.86T + 23T^{2} \) |
| 29 | \( 1 + 1.87T + 29T^{2} \) |
| 31 | \( 1 - 1.35T + 31T^{2} \) |
| 37 | \( 1 + 1.13T + 37T^{2} \) |
| 41 | \( 1 + 0.521T + 41T^{2} \) |
| 43 | \( 1 + 2.07T + 43T^{2} \) |
| 47 | \( 1 + 0.616T + 47T^{2} \) |
| 53 | \( 1 - 8.64T + 53T^{2} \) |
| 59 | \( 1 + 0.581T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 7.57T + 79T^{2} \) |
| 83 | \( 1 + 0.913T + 83T^{2} \) |
| 89 | \( 1 + 3.68T + 89T^{2} \) |
| 97 | \( 1 + 3.24T + 97T^{2} \) |
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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
−9.710060309839149296869419431504, −8.348859029676675837113472960543, −7.933840268319171895939908859609, −7.07500836463896683270480969801, −6.14135430916937952292622220633, −5.25675938605378852465656709251, −3.96849129225057838099056681716, −3.87586918025555121834895782523, −2.88689898511333621198634945498, −1.82830299085773581438133271029,
1.82830299085773581438133271029, 2.88689898511333621198634945498, 3.87586918025555121834895782523, 3.96849129225057838099056681716, 5.25675938605378852465656709251, 6.14135430916937952292622220633, 7.07500836463896683270480969801, 7.933840268319171895939908859609, 8.348859029676675837113472960543, 9.710060309839149296869419431504
