| L(s) = 1 | − 4.46·5-s − 0.464·13-s − 7.92·17-s + 14.9·25-s − 8.46·29-s − 11.3·37-s + 10·41-s − 7·49-s − 14·53-s + 5.39·61-s + 2.07·65-s − 10.8·73-s + 35.3·85-s + 8.85·89-s + 18·97-s + 2·101-s + 20.3·109-s − 6.85·113-s + ⋯ |
| L(s) = 1 | − 1.99·5-s − 0.128·13-s − 1.92·17-s + 2.98·25-s − 1.57·29-s − 1.87·37-s + 1.56·41-s − 49-s − 1.92·53-s + 0.690·61-s + 0.256·65-s − 1.27·73-s + 3.83·85-s + 0.938·89-s + 1.82·97-s + 0.199·101-s + 1.94·109-s − 0.644·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5137331591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5137331591\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 4.46T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 0.464T + 13T^{2} \) |
| 17 | \( 1 + 7.92T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 8.46T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 14T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 5.39T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 8.85T + 89T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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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
−8.156612511271434346526020858050, −7.47097474835230567969543299763, −7.00686009611450678944376664071, −6.22022506250851922997472385938, −5.01331836638713343789264024931, −4.44494178778729488061745237176, −3.77906035919093832763527125450, −3.07409752224957339324227664587, −1.90448245573759145325687976885, −0.37288790028450785323453949242,
0.37288790028450785323453949242, 1.90448245573759145325687976885, 3.07409752224957339324227664587, 3.77906035919093832763527125450, 4.44494178778729488061745237176, 5.01331836638713343789264024931, 6.22022506250851922997472385938, 7.00686009611450678944376664071, 7.47097474835230567969543299763, 8.156612511271434346526020858050
