L(s) = 1 | + 128·4-s − 1.76e3·7-s + 1.26e4·13-s + 5.74e4·19-s − 1.56e5·25-s − 2.25e5·28-s − 6.62e5·31-s − 2.79e5·37-s + 6.25e5·43-s + 8.23e5·49-s + 1.61e6·52-s − 1.99e6·61-s − 2.09e6·64-s − 4.44e6·67-s + 1.00e7·73-s + 7.35e6·76-s − 9.03e6·79-s − 2.22e7·91-s + 1.75e7·97-s − 2.00e7·100-s − 4.38e7·103-s + 5.34e7·109-s + 1.94e7·121-s − 8.48e7·124-s + 127-s + 131-s − 1.01e8·133-s + ⋯ |
L(s) = 1 | + 4-s − 1.94·7-s + 1.59·13-s + 1.92·19-s − 2·25-s − 1.94·28-s − 3.99·31-s − 0.907·37-s + 1.20·43-s + 49-s + 1.59·52-s − 1.12·61-s − 64-s − 1.80·67-s + 3.03·73-s + 1.92·76-s − 2.06·79-s − 3.09·91-s + 1.94·97-s − 2·100-s − 3.95·103-s + 3.95·109-s + 121-s − 3.99·124-s − 3.73·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(1.205701714\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.205701714\) |
\(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 12605 T + p^{7} T^{2} \) |
good | 2 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 508 T + p^{7} T^{2} )( 1 + 1255 T + p^{7} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 43091 T + p^{7} T^{2} )( 1 - 14357 T + p^{7} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 331387 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 335663 T + p^{7} T^{2} )( 1 + 615373 T + p^{7} T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 1035224 T + p^{7} T^{2} )( 1 + 409495 T + p^{7} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 1537199 T + p^{7} T^{2} )( 1 + 3535546 T + p^{7} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 385072 T + p^{7} T^{2} )( 1 + 4058455 T + p^{7} T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 5038001 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4517617 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 12245198 T + p^{7} T^{2} )( 1 - 5276357 T + p^{7} T^{2} ) \) |
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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
−12.61593972616659772037447655996, −11.73447576609460095821174292080, −11.47720137033973757827720181597, −10.81827676207360587660495376622, −10.50323342657472674293163692824, −9.512109378067277170689083727635, −9.458264031270819351117623266355, −8.925990233820996665478086720144, −7.927576317124303856636911842824, −7.17809003605299103764493011803, −7.16398485061566486905513472860, −6.12551683197131201048524442305, −5.95312127751398422826120290221, −5.37978071273508262860224714024, −3.97049350825627285084866584456, −3.42219124083318195414951729401, −3.18649607313584200768438525370, −2.05015100499069875518352988895, −1.45960808699802255086680717662, −0.30890239167143525101681596319,
0.30890239167143525101681596319, 1.45960808699802255086680717662, 2.05015100499069875518352988895, 3.18649607313584200768438525370, 3.42219124083318195414951729401, 3.97049350825627285084866584456, 5.37978071273508262860224714024, 5.95312127751398422826120290221, 6.12551683197131201048524442305, 7.16398485061566486905513472860, 7.17809003605299103764493011803, 7.927576317124303856636911842824, 8.925990233820996665478086720144, 9.458264031270819351117623266355, 9.512109378067277170689083727635, 10.50323342657472674293163692824, 10.81827676207360587660495376622, 11.47720137033973757827720181597, 11.73447576609460095821174292080, 12.61593972616659772037447655996