| L(s) = 1 | − 2·9-s + 16·13-s − 22·16-s + 88·19-s − 176·25-s + 64·43-s + 148·49-s + 408·67-s + 30·81-s − 168·103-s − 32·117-s − 468·121-s + 127-s + 131-s + 137-s + 139-s + 44·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 872·169-s − 176·171-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2/9·9-s + 1.23·13-s − 1.37·16-s + 4.63·19-s − 7.03·25-s + 1.48·43-s + 3.02·49-s + 6.08·67-s + 0.370·81-s − 1.63·103-s − 0.273·117-s − 3.86·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.305·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 5.15·169-s − 1.02·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.652958185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.652958185\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 2 T^{2} - 26 T^{4} + 2 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | \( 1 - 228 T^{2} + 566 p^{2} T^{4} - 228 p^{4} T^{6} + p^{8} T^{8} \) |
| good | 2 | \( ( 1 + 11 T^{4} + p^{8} T^{8} )^{2} \) |
| 5 | \( ( 1 + 44 T^{2} + p^{4} T^{4} )^{4} \) |
| 7 | \( ( 1 - 74 T^{2} + 2622 T^{4} - 74 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 + 234 T^{2} + 41270 T^{4} + 234 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 4 T + 258 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 - 22 T + 654 T^{2} - 22 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 + 1602 T^{2} + 1197734 T^{4} + 1602 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 + 1048 T^{2} + 1580274 T^{4} + 1048 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 1802 T^{2} + 1723902 T^{4} - 1802 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 2804 T^{2} + 5683602 T^{4} - 2804 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 + 3324 T^{2} + 3350 p^{2} T^{4} + 3324 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 - 16 T + 3678 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 47 | \( ( 1 - 5472 T^{2} + 14845934 T^{4} - 5472 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 6040 T^{2} + 19941246 T^{4} - 6040 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 - 12292 T^{2} + 61496982 T^{4} - 12292 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 3812 T^{2} + 27513522 T^{4} - 3812 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 102 T + 11558 T^{2} - 102 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 + 19450 T^{2} + 145315638 T^{4} + 19450 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 + 956 T^{2} + 47541702 T^{4} + 956 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 1514 T^{2} + 72685086 T^{4} - 1514 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 - 22372 T^{2} + 217838742 T^{4} - 22372 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 2016 T^{2} - 61744210 T^{4} - 2016 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 35852 T^{2} + 498399702 T^{4} - 35852 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.33089936623867053031050888675, −7.15029525909461996854367345643, −6.94426016571440627665468525846, −6.77335490906774580941108126294, −6.61396316506521191868808097843, −6.09781472550798687940720749481, −6.03345917406229548035967412572, −5.92491578361191086277469898620, −5.53854526493286376500788493863, −5.51266334846014138555761020862, −5.49773853157214704726905658884, −5.34797083849669880743686935305, −4.81559935684394294731331414185, −4.69401540329311389744589657593, −4.12059314360278516823453543206, −4.01160175016841556594232961109, −3.80329767339686070086888289863, −3.72016093841469492290259173737, −3.58181168726986202142985212459, −3.12094832820564294532674474154, −2.49518000828973433702997610719, −2.41907705094827182736284859225, −2.06802091811646304991144507318, −1.49649850718889595580400374123, −0.76836217272854838477194932713,
0.76836217272854838477194932713, 1.49649850718889595580400374123, 2.06802091811646304991144507318, 2.41907705094827182736284859225, 2.49518000828973433702997610719, 3.12094832820564294532674474154, 3.58181168726986202142985212459, 3.72016093841469492290259173737, 3.80329767339686070086888289863, 4.01160175016841556594232961109, 4.12059314360278516823453543206, 4.69401540329311389744589657593, 4.81559935684394294731331414185, 5.34797083849669880743686935305, 5.49773853157214704726905658884, 5.51266334846014138555761020862, 5.53854526493286376500788493863, 5.92491578361191086277469898620, 6.03345917406229548035967412572, 6.09781472550798687940720749481, 6.61396316506521191868808097843, 6.77335490906774580941108126294, 6.94426016571440627665468525846, 7.15029525909461996854367345643, 7.33089936623867053031050888675
Plot not available for L-functions of degree greater than 10.
