| L(s) = 1 | + 4.24·5-s + 7-s − 5.74·11-s + 6.06·13-s + 2.56·17-s + 7.74·19-s − 1.67·23-s + 12.9·25-s + 5.24·29-s + 5.24·31-s + 4.24·35-s − 7.98·37-s + 2.24·41-s − 5.92·43-s − 6.62·47-s + 49-s + 2.85·53-s − 24.3·55-s − 7.48·59-s − 12.2·61-s + 25.7·65-s + 8.16·67-s − 2.32·71-s − 1.00·73-s − 5.74·77-s − 7.59·79-s + 13.9·83-s + ⋯ |
| L(s) = 1 | + 1.89·5-s + 0.377·7-s − 1.73·11-s + 1.68·13-s + 0.621·17-s + 1.77·19-s − 0.350·23-s + 2.59·25-s + 0.973·29-s + 0.941·31-s + 0.716·35-s − 1.31·37-s + 0.349·41-s − 0.902·43-s − 0.966·47-s + 0.142·49-s + 0.392·53-s − 3.28·55-s − 0.973·59-s − 1.56·61-s + 3.18·65-s + 0.996·67-s − 0.275·71-s − 0.117·73-s − 0.654·77-s − 0.855·79-s + 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.439647309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.439647309\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 4.24T + 5T^{2} \) |
| 11 | \( 1 + 5.74T + 11T^{2} \) |
| 13 | \( 1 - 6.06T + 13T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 - 7.74T + 19T^{2} \) |
| 23 | \( 1 + 1.67T + 23T^{2} \) |
| 29 | \( 1 - 5.24T + 29T^{2} \) |
| 31 | \( 1 - 5.24T + 31T^{2} \) |
| 37 | \( 1 + 7.98T + 37T^{2} \) |
| 41 | \( 1 - 2.24T + 41T^{2} \) |
| 43 | \( 1 + 5.92T + 43T^{2} \) |
| 47 | \( 1 + 6.62T + 47T^{2} \) |
| 53 | \( 1 - 2.85T + 53T^{2} \) |
| 59 | \( 1 + 7.48T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 8.16T + 67T^{2} \) |
| 71 | \( 1 + 2.32T + 71T^{2} \) |
| 73 | \( 1 + 1.00T + 73T^{2} \) |
| 79 | \( 1 + 7.59T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 3.61T + 97T^{2} \) |
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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
−8.189189633572324797836597396452, −7.39603678761350292246518089443, −6.42431155203979807788355274914, −5.88557548163400749815304761647, −5.27213260918953613100976377152, −4.83408724357781085213142085207, −3.31824731943966133487453945493, −2.79699565962237788890786155643, −1.75342707279247963894325655693, −1.07189988440536400761046579058,
1.07189988440536400761046579058, 1.75342707279247963894325655693, 2.79699565962237788890786155643, 3.31824731943966133487453945493, 4.83408724357781085213142085207, 5.27213260918953613100976377152, 5.88557548163400749815304761647, 6.42431155203979807788355274914, 7.39603678761350292246518089443, 8.189189633572324797836597396452
