| L(s) = 1 | + 38·4-s − 806·7-s − 1.92e3·13-s − 2.65e3·16-s + 1.60e4·19-s + 1.36e4·25-s − 3.06e4·28-s + 9.77e4·31-s + 4.83e4·37-s − 1.21e5·43-s + 2.51e5·49-s − 7.30e4·52-s + 5.45e5·61-s − 2.56e5·64-s − 1.71e5·67-s − 3.05e5·73-s + 6.09e5·76-s − 1.48e5·79-s + 1.54e6·91-s − 2.39e6·97-s + 5.17e5·100-s + 2.34e6·103-s + 3.77e5·109-s + 2.13e6·112-s + 1.29e6·121-s + 3.71e6·124-s + 127-s + ⋯ |
| L(s) = 1 | + 0.593·4-s − 2.34·7-s − 0.874·13-s − 0.647·16-s + 2.33·19-s + 0.871·25-s − 1.39·28-s + 3.27·31-s + 0.954·37-s − 1.52·43-s + 2.14·49-s − 0.519·52-s + 2.40·61-s − 0.978·64-s − 0.569·67-s − 0.785·73-s + 1.38·76-s − 0.300·79-s + 2.05·91-s − 2.62·97-s + 0.517·100-s + 2.14·103-s + 0.291·109-s + 1.52·112-s + 0.731·121-s + 1.94·124-s − 5.49·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.528183372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.528183372\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 19 p T^{2} + p^{12} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2722 p T^{2} + p^{12} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 403 T + p^{6} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 1296362 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 961 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 44262502 T^{2} + p^{12} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8021 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 183578618 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1183562642 T^{2} + p^{12} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 48854 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 24167 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 4491564482 T^{2} + p^{12} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 1414 p T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 20829456218 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25385336498 T^{2} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 74846310122 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 272999 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 85579 T + p^{6} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 139404996002 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 152737 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 74059 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 644553830738 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 430933822918 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1197313 T + p^{6} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.43328849811624530887720060325, −15.70354160908033361893694124742, −15.48528904324243096361972788896, −14.48987751077665638330503509048, −13.61115121788217362920383645640, −13.31608998005064072822578027533, −12.51230793523069452017677542745, −11.87650182552932993478558746076, −11.43634296735836123638826621455, −10.01190516188246852502575111419, −9.986836991501613221822444285694, −9.320413258663293903084936825604, −8.203161728507335186059484202552, −7.03267054245268434675094644345, −6.76987967388284828238587835056, −5.90714014983130655915811795132, −4.71137393850273725566382844012, −3.17000916963960605459453982421, −2.76805068737731192224677330883, −0.70185214480232132091393061711,
0.70185214480232132091393061711, 2.76805068737731192224677330883, 3.17000916963960605459453982421, 4.71137393850273725566382844012, 5.90714014983130655915811795132, 6.76987967388284828238587835056, 7.03267054245268434675094644345, 8.203161728507335186059484202552, 9.320413258663293903084936825604, 9.986836991501613221822444285694, 10.01190516188246852502575111419, 11.43634296735836123638826621455, 11.87650182552932993478558746076, 12.51230793523069452017677542745, 13.31608998005064072822578027533, 13.61115121788217362920383645640, 14.48987751077665638330503509048, 15.48528904324243096361972788896, 15.70354160908033361893694124742, 16.43328849811624530887720060325
