| L(s) = 1 | − 2.31·2-s + 3.36·4-s + 0.683·7-s − 3.16·8-s − 3.68·11-s + 4.43·13-s − 1.58·14-s + 0.593·16-s + 17-s − 1.03·19-s + 8.52·22-s + 4.52·23-s − 10.2·26-s + 2.30·28-s + 3.69·29-s − 10.8·31-s + 4.94·32-s − 2.31·34-s + 0.308·37-s + 2.40·38-s + 6.15·41-s + 7.88·43-s − 12.3·44-s − 10.4·46-s − 4.43·47-s − 6.53·49-s + 14.9·52-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 1.68·4-s + 0.258·7-s − 1.11·8-s − 1.10·11-s + 1.23·13-s − 0.423·14-s + 0.148·16-s + 0.242·17-s − 0.238·19-s + 1.81·22-s + 0.943·23-s − 2.01·26-s + 0.434·28-s + 0.685·29-s − 1.95·31-s + 0.874·32-s − 0.397·34-s + 0.0507·37-s + 0.390·38-s + 0.960·41-s + 1.20·43-s − 1.86·44-s − 1.54·46-s − 0.647·47-s − 0.933·49-s + 2.07·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7762459471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7762459471\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + 2.31T + 2T^{2} \) |
| 7 | \( 1 - 0.683T + 7T^{2} \) |
| 11 | \( 1 + 3.68T + 11T^{2} \) |
| 13 | \( 1 - 4.43T + 13T^{2} \) |
| 19 | \( 1 + 1.03T + 19T^{2} \) |
| 23 | \( 1 - 4.52T + 23T^{2} \) |
| 29 | \( 1 - 3.69T + 29T^{2} \) |
| 31 | \( 1 + 10.8T + 31T^{2} \) |
| 37 | \( 1 - 0.308T + 37T^{2} \) |
| 41 | \( 1 - 6.15T + 41T^{2} \) |
| 43 | \( 1 - 7.88T + 43T^{2} \) |
| 47 | \( 1 + 4.43T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 2T + 59T^{2} \) |
| 61 | \( 1 - 9.94T + 61T^{2} \) |
| 67 | \( 1 - 9.16T + 67T^{2} \) |
| 71 | \( 1 - 9.37T + 71T^{2} \) |
| 73 | \( 1 + 2.26T + 73T^{2} \) |
| 79 | \( 1 - 7.42T + 79T^{2} \) |
| 83 | \( 1 - 8.92T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−8.399312472115025014031573735527, −8.031623318067547917357286814059, −7.30228877160264245176305145483, −6.57253599896956989530308281551, −5.70293881582890574221269340396, −4.83802423080120728577736826120, −3.63191966119162799910998663584, −2.61547551428463010141248765546, −1.67172703414707151447265011646, −0.66956078201007422413541844219,
0.66956078201007422413541844219, 1.67172703414707151447265011646, 2.61547551428463010141248765546, 3.63191966119162799910998663584, 4.83802423080120728577736826120, 5.70293881582890574221269340396, 6.57253599896956989530308281551, 7.30228877160264245176305145483, 8.031623318067547917357286814059, 8.399312472115025014031573735527
