| L(s) = 1 | + 2.52·2-s + 4.39·4-s − 4.92·7-s + 6.05·8-s + 1.13·11-s + 4·13-s − 12.4·14-s + 6.52·16-s + 6.79·17-s − 19-s + 2.86·22-s − 1.92·23-s + 10.1·26-s − 21.6·28-s + 5·29-s + 5.13·31-s + 4.39·32-s + 17.1·34-s + 9.05·37-s − 2.52·38-s − 3.86·41-s + 4·43-s + 4.98·44-s − 4.86·46-s − 4.26·47-s + 17.2·49-s + 17.5·52-s + ⋯ |
| L(s) = 1 | + 1.78·2-s + 2.19·4-s − 1.86·7-s + 2.14·8-s + 0.341·11-s + 1.10·13-s − 3.32·14-s + 1.63·16-s + 1.64·17-s − 0.229·19-s + 0.611·22-s − 0.401·23-s + 1.98·26-s − 4.09·28-s + 0.928·29-s + 0.922·31-s + 0.777·32-s + 2.94·34-s + 1.48·37-s − 0.410·38-s − 0.603·41-s + 0.609·43-s + 0.751·44-s − 0.717·46-s − 0.622·47-s + 2.46·49-s + 2.43·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.565351413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.565351413\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.52T + 2T^{2} \) |
| 7 | \( 1 + 4.92T + 7T^{2} \) |
| 11 | \( 1 - 1.13T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 6.79T + 17T^{2} \) |
| 23 | \( 1 + 1.92T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 5.13T + 31T^{2} \) |
| 37 | \( 1 - 9.05T + 37T^{2} \) |
| 41 | \( 1 + 3.86T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 4.26T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 9.19T + 59T^{2} \) |
| 61 | \( 1 - 0.733T + 61T^{2} \) |
| 67 | \( 1 - 4.86T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + 0.866T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 9.32T + 97T^{2} \) |
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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
−8.197346088735846038674474210800, −7.26161459876849882633847052211, −6.49367529920704214041316177307, −6.10039755034734567915224268915, −5.59772730884063189680078578888, −4.47072332759465064448197358266, −3.73657120223801140246363342094, −3.23202477100561047745649211213, −2.55568794204093550042405489211, −1.06135813064083074717092340932,
1.06135813064083074717092340932, 2.55568794204093550042405489211, 3.23202477100561047745649211213, 3.73657120223801140246363342094, 4.47072332759465064448197358266, 5.59772730884063189680078578888, 6.10039755034734567915224268915, 6.49367529920704214041316177307, 7.26161459876849882633847052211, 8.197346088735846038674474210800
