| L(s) = 1 | − 27·3-s + 64·4-s − 286·7-s + 729·9-s − 1.72e3·12-s + 506·13-s + 4.09e3·16-s − 1.05e4·19-s + 7.72e3·21-s + 1.56e4·25-s − 1.96e4·27-s − 1.83e4·28-s + 3.52e4·31-s + 4.66e4·36-s − 8.92e4·37-s − 1.36e4·39-s + 1.11e5·43-s − 1.10e5·48-s − 3.58e4·49-s + 3.23e4·52-s + 2.85e5·57-s − 4.20e5·61-s − 2.08e5·63-s + 2.62e5·64-s + 1.72e5·67-s + 6.38e5·73-s − 4.21e5·75-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s − 0.833·7-s + 9-s − 12-s + 0.230·13-s + 16-s − 1.54·19-s + 0.833·21-s + 25-s − 27-s − 0.833·28-s + 1.18·31-s + 36-s − 1.76·37-s − 0.230·39-s + 1.40·43-s − 48-s − 0.304·49-s + 0.230·52-s + 1.54·57-s − 1.85·61-s − 0.833·63-s + 64-s + 0.574·67-s + 1.64·73-s − 75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.8025269875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8025269875\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{3} T \) |
| good | 2 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 5 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 7 | \( 1 + 286 T + p^{6} T^{2} \) |
| 11 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 13 | \( 1 - 506 T + p^{6} T^{2} \) |
| 17 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 19 | \( 1 + 10582 T + p^{6} T^{2} \) |
| 23 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 29 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 31 | \( 1 - 35282 T + p^{6} T^{2} \) |
| 37 | \( 1 + 89206 T + p^{6} T^{2} \) |
| 41 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 43 | \( 1 - 111386 T + p^{6} T^{2} \) |
| 47 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 53 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 59 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 61 | \( 1 + 420838 T + p^{6} T^{2} \) |
| 67 | \( 1 - 172874 T + p^{6} T^{2} \) |
| 71 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 73 | \( 1 - 638066 T + p^{6} T^{2} \) |
| 79 | \( 1 + 204622 T + p^{6} T^{2} \) |
| 83 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 89 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 97 | \( 1 + 56446 T + p^{6} T^{2} \) |
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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
−25.91719154013093344414424320834, −24.31625596819603239877660545934, −22.81040584989870861590484456027, −21.14553963209100554789272819114, −19.16808984714065757854822219386, −16.98315802839373498452012617625, −15.64955611967823946795560751404, −12.47129821992859305641569206852, −10.64402516981818544954724445589, −6.51020658700370227865993577833,
6.51020658700370227865993577833, 10.64402516981818544954724445589, 12.47129821992859305641569206852, 15.64955611967823946795560751404, 16.98315802839373498452012617625, 19.16808984714065757854822219386, 21.14553963209100554789272819114, 22.81040584989870861590484456027, 24.31625596819603239877660545934, 25.91719154013093344414424320834
