| L(s) = 1 | + 0.526·2-s − 2.50·3-s − 1.72·4-s − 0.0684·5-s − 1.32·6-s − 3.53·7-s − 1.96·8-s + 3.29·9-s − 0.0360·10-s − 3.28·11-s + 4.32·12-s − 4.46·13-s − 1.86·14-s + 0.171·15-s + 2.41·16-s − 6.41·17-s + 1.73·18-s − 1.03·19-s + 0.117·20-s + 8.86·21-s − 1.72·22-s + 4.56·23-s + 4.91·24-s − 4.99·25-s − 2.34·26-s − 0.729·27-s + 6.08·28-s + ⋯ |
| L(s) = 1 | + 0.372·2-s − 1.44·3-s − 0.861·4-s − 0.0306·5-s − 0.539·6-s − 1.33·7-s − 0.693·8-s + 1.09·9-s − 0.0113·10-s − 0.990·11-s + 1.24·12-s − 1.23·13-s − 0.497·14-s + 0.0443·15-s + 0.603·16-s − 1.55·17-s + 0.408·18-s − 0.236·19-s + 0.0263·20-s + 1.93·21-s − 0.368·22-s + 0.951·23-s + 1.00·24-s − 0.999·25-s − 0.460·26-s − 0.140·27-s + 1.15·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08980053017\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08980053017\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 \) |
| good | 2 | \( 1 - 0.526T + 2T^{2} \) |
| 3 | \( 1 + 2.50T + 3T^{2} \) |
| 5 | \( 1 + 0.0684T + 5T^{2} \) |
| 7 | \( 1 + 3.53T + 7T^{2} \) |
| 11 | \( 1 + 3.28T + 11T^{2} \) |
| 13 | \( 1 + 4.46T + 13T^{2} \) |
| 17 | \( 1 + 6.41T + 17T^{2} \) |
| 19 | \( 1 + 1.03T + 19T^{2} \) |
| 23 | \( 1 - 4.56T + 23T^{2} \) |
| 29 | \( 1 + 6.88T + 29T^{2} \) |
| 31 | \( 1 + 3.11T + 31T^{2} \) |
| 41 | \( 1 - 0.132T + 41T^{2} \) |
| 43 | \( 1 - 1.05T + 43T^{2} \) |
| 47 | \( 1 - 6.28T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 0.295T + 59T^{2} \) |
| 61 | \( 1 - 0.197T + 61T^{2} \) |
| 67 | \( 1 + 8.35T + 67T^{2} \) |
| 71 | \( 1 - 5.41T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 5.50T + 83T^{2} \) |
| 89 | \( 1 - 4.93T + 89T^{2} \) |
| 97 | \( 1 - 5.81T + 97T^{2} \) |
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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
−9.548263542342241366666241192673, −9.165099774860214612792343344450, −7.79473696831766621268356100978, −6.85464967601703927523705420269, −6.13322531691578023820210965799, −5.33177999234003609415853367955, −4.76771251244674637056047693228, −3.76008466374010197942538667782, −2.52423755742608452672634116798, −0.20505265866272653017409021063,
0.20505265866272653017409021063, 2.52423755742608452672634116798, 3.76008466374010197942538667782, 4.76771251244674637056047693228, 5.33177999234003609415853367955, 6.13322531691578023820210965799, 6.85464967601703927523705420269, 7.79473696831766621268356100978, 9.165099774860214612792343344450, 9.548263542342241366666241192673
