| L(s) = 1 | − 6·5-s + 9·9-s + 10·13-s − 30·17-s + 11·25-s + 42·29-s − 70·37-s + 18·41-s − 54·45-s + 49·49-s + 90·53-s − 22·61-s − 60·65-s − 110·73-s + 81·81-s + 180·85-s − 78·89-s + 130·97-s − 198·101-s − 182·109-s − 30·113-s + 90·117-s + ⋯ |
| L(s) = 1 | − 6/5·5-s + 9-s + 0.769·13-s − 1.76·17-s + 0.439·25-s + 1.44·29-s − 1.89·37-s + 0.439·41-s − 6/5·45-s + 49-s + 1.69·53-s − 0.360·61-s − 0.923·65-s − 1.50·73-s + 81-s + 2.11·85-s − 0.876·89-s + 1.34·97-s − 1.96·101-s − 1.66·109-s − 0.265·113-s + 0.769·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7392159185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7392159185\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 5 | \( 1 + 6 T + p^{2} T^{2} \) |
| 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 - 10 T + p^{2} T^{2} \) |
| 17 | \( 1 + 30 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 - 42 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 + 70 T + p^{2} T^{2} \) |
| 41 | \( 1 - 18 T + p^{2} T^{2} \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 - 90 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 + 22 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 110 T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 + 78 T + p^{2} T^{2} \) |
| 97 | \( 1 - 130 T + p^{2} T^{2} \) |
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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
−19.06051373579605759802271786117, −17.82947447057657457193643002853, −15.98031784304075915422898153882, −15.39450427933260879596695517146, −13.47743853960170012065633600349, −12.04193928944153810810388327001, −10.66144251061067377468104379504, −8.616071000238428753761934055988, −6.97817398550361802636417654463, −4.20532042858809199731938716994,
4.20532042858809199731938716994, 6.97817398550361802636417654463, 8.616071000238428753761934055988, 10.66144251061067377468104379504, 12.04193928944153810810388327001, 13.47743853960170012065633600349, 15.39450427933260879596695517146, 15.98031784304075915422898153882, 17.82947447057657457193643002853, 19.06051373579605759802271786117
