| L(s) = 1 | − 3-s − 1.19·5-s − 1.77·7-s + 9-s + 5.47·11-s − 4.71·13-s + 1.19·15-s + 4.92·17-s + 8.56·19-s + 1.77·21-s + 4.04·23-s − 3.57·25-s − 27-s + 9.38·29-s + 0.0932·31-s − 5.47·33-s + 2.11·35-s − 1.65·37-s + 4.71·39-s + 0.208·41-s − 6.12·43-s − 1.19·45-s + 5.00·47-s − 3.85·49-s − 4.92·51-s − 8.99·53-s − 6.54·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.534·5-s − 0.669·7-s + 0.333·9-s + 1.65·11-s − 1.30·13-s + 0.308·15-s + 1.19·17-s + 1.96·19-s + 0.386·21-s + 0.843·23-s − 0.714·25-s − 0.192·27-s + 1.74·29-s + 0.0167·31-s − 0.952·33-s + 0.358·35-s − 0.272·37-s + 0.755·39-s + 0.0325·41-s − 0.934·43-s − 0.178·45-s + 0.730·47-s − 0.551·49-s − 0.689·51-s − 1.23·53-s − 0.882·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.517224148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.517224148\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 1.19T + 5T^{2} \) |
| 7 | \( 1 + 1.77T + 7T^{2} \) |
| 11 | \( 1 - 5.47T + 11T^{2} \) |
| 13 | \( 1 + 4.71T + 13T^{2} \) |
| 17 | \( 1 - 4.92T + 17T^{2} \) |
| 19 | \( 1 - 8.56T + 19T^{2} \) |
| 23 | \( 1 - 4.04T + 23T^{2} \) |
| 29 | \( 1 - 9.38T + 29T^{2} \) |
| 31 | \( 1 - 0.0932T + 31T^{2} \) |
| 37 | \( 1 + 1.65T + 37T^{2} \) |
| 41 | \( 1 - 0.208T + 41T^{2} \) |
| 43 | \( 1 + 6.12T + 43T^{2} \) |
| 47 | \( 1 - 5.00T + 47T^{2} \) |
| 53 | \( 1 + 8.99T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 9.91T + 61T^{2} \) |
| 67 | \( 1 + 4.73T + 67T^{2} \) |
| 71 | \( 1 - 6.89T + 71T^{2} \) |
| 73 | \( 1 - 2.90T + 73T^{2} \) |
| 79 | \( 1 + 2.36T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 7.84T + 89T^{2} \) |
| 97 | \( 1 - 5.91T + 97T^{2} \) |
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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
−7.61951476057627774949593531268, −7.09201833712064145045861072401, −6.57662446710830518436482603686, −5.73654290231156888158134274066, −5.05118246962826716172880406292, −4.34546438439799322149975870338, −3.42487894778293536244251293215, −2.96843304460583425392424748585, −1.47507639130570112193418723037, −0.68285326389791053747010875196,
0.68285326389791053747010875196, 1.47507639130570112193418723037, 2.96843304460583425392424748585, 3.42487894778293536244251293215, 4.34546438439799322149975870338, 5.05118246962826716172880406292, 5.73654290231156888158134274066, 6.57662446710830518436482603686, 7.09201833712064145045861072401, 7.61951476057627774949593531268
