| L(s) = 1 | − 6.56e3·3-s + 1.30e5·5-s + 1.48e7·7-s + 4.30e7·9-s + 8.45e8·11-s + 1.75e9·13-s − 8.59e8·15-s − 4.71e7·17-s + 5.69e10·19-s − 9.74e10·21-s + 3.71e11·23-s − 7.45e11·25-s − 2.82e11·27-s − 3.68e12·29-s + 5.47e12·31-s − 5.54e12·33-s + 1.94e12·35-s − 5.44e12·37-s − 1.14e13·39-s + 2.97e13·41-s − 9.84e13·43-s + 5.63e12·45-s − 1.07e14·47-s − 1.22e13·49-s + 3.09e11·51-s + 6.26e14·53-s + 1.10e14·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.149·5-s + 0.973·7-s + 1/3·9-s + 1.18·11-s + 0.595·13-s − 0.0865·15-s − 0.00163·17-s + 0.769·19-s − 0.562·21-s + 0.988·23-s − 0.977·25-s − 0.192·27-s − 1.36·29-s + 1.15·31-s − 0.686·33-s + 0.145·35-s − 0.254·37-s − 0.343·39-s + 0.582·41-s − 1.28·43-s + 0.0499·45-s − 0.660·47-s − 0.0524·49-s + 0.000946·51-s + 1.38·53-s + 0.178·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(2.481761647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.481761647\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{8} T \) |
| good | 5 | \( 1 - 5238 p^{2} T + p^{17} T^{2} \) |
| 7 | \( 1 - 2120968 p T + p^{17} T^{2} \) |
| 11 | \( 1 - 845469684 T + p^{17} T^{2} \) |
| 13 | \( 1 - 134724230 p T + p^{17} T^{2} \) |
| 17 | \( 1 + 47147886 T + p^{17} T^{2} \) |
| 19 | \( 1 - 56973573100 T + p^{17} T^{2} \) |
| 23 | \( 1 - 371395374696 T + p^{17} T^{2} \) |
| 29 | \( 1 + 3681168479586 T + p^{17} T^{2} \) |
| 31 | \( 1 - 5479889229856 T + p^{17} T^{2} \) |
| 37 | \( 1 + 5446958938138 T + p^{17} T^{2} \) |
| 41 | \( 1 - 29773337634090 T + p^{17} T^{2} \) |
| 43 | \( 1 + 98485895466284 T + p^{17} T^{2} \) |
| 47 | \( 1 + 107861800207536 T + p^{17} T^{2} \) |
| 53 | \( 1 - 626472886328118 T + p^{17} T^{2} \) |
| 59 | \( 1 - 1260971066668356 T + p^{17} T^{2} \) |
| 61 | \( 1 + 956343149707138 T + p^{17} T^{2} \) |
| 67 | \( 1 - 5519389511567164 T + p^{17} T^{2} \) |
| 71 | \( 1 + 9303053873586120 T + p^{17} T^{2} \) |
| 73 | \( 1 - 3692590926453962 T + p^{17} T^{2} \) |
| 79 | \( 1 - 2597720120860912 T + p^{17} T^{2} \) |
| 83 | \( 1 + 26266742515599444 T + p^{17} T^{2} \) |
| 89 | \( 1 - 63717157489864410 T + p^{17} T^{2} \) |
| 97 | \( 1 + 809885530989406 p T + p^{17} T^{2} \) |
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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
−11.75384157845545154706064785375, −11.27081262304980498745042231241, −9.845083227289673124551920597512, −8.619971715490553627287081029436, −7.27259779288657934777032011358, −6.03679589067746262089470552415, −4.88964244427636343051326498003, −3.65247350613669375967697392599, −1.78520847956283966736514753796, −0.871976413664304181559853739562,
0.871976413664304181559853739562, 1.78520847956283966736514753796, 3.65247350613669375967697392599, 4.88964244427636343051326498003, 6.03679589067746262089470552415, 7.27259779288657934777032011358, 8.619971715490553627287081029436, 9.845083227289673124551920597512, 11.27081262304980498745042231241, 11.75384157845545154706064785375
