| L(s) = 1 | + 6·4-s − 8·13-s + 19·16-s + 32·47-s − 48·52-s + 36·64-s + 48·67-s − 81-s + 40·89-s − 24·101-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 192·188-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 3·4-s − 2.21·13-s + 19/4·16-s + 4.66·47-s − 6.65·52-s + 9/2·64-s + 5.86·67-s − 1/9·81-s + 4.23·89-s − 2.38·101-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 14.0·188-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.076695451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.076695451\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 17 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 - 3 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^3$ | \( 1 - 94 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 206 T^{4} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 + 194 T^{4} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 - 2638 T^{4} + p^{4} T^{8} \) |
| 41 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )( 1 + 80 T^{2} + p^{2} T^{4} ) \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^3$ | \( 1 - 4078 T^{4} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 71 | $C_2^3$ | \( 1 - 10078 T^{4} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 + 7682 T^{4} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 97 | $C_2^3$ | \( 1 - 14974 T^{4} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.24855481526897250176318709932, −7.09446953773647332066480738063, −6.82774272324287118620274547717, −6.65067942891167034305187767781, −6.49149435301475766187675589905, −6.33353520234343961750241681204, −5.94583115840277757570483074124, −5.71789386427427522169938518220, −5.41011017483418362932083092950, −5.40548779991955172021917267478, −5.02926382336883831668377067466, −4.74623200455836089600652149511, −4.54294338165024402937305069733, −3.99799860331990027746249921808, −3.77751178235424578980270167458, −3.65216790123326349616584524263, −3.31417160239880765780433016567, −2.75412167457608938169415456766, −2.50622774027584122798337491566, −2.40105704805085328042931097410, −2.36943668241269769636834931132, −2.06619823248521273630654774686, −1.53927791480247816916690213492, −1.04011391364982053581673139782, −0.60823112969553030483359783508,
0.60823112969553030483359783508, 1.04011391364982053581673139782, 1.53927791480247816916690213492, 2.06619823248521273630654774686, 2.36943668241269769636834931132, 2.40105704805085328042931097410, 2.50622774027584122798337491566, 2.75412167457608938169415456766, 3.31417160239880765780433016567, 3.65216790123326349616584524263, 3.77751178235424578980270167458, 3.99799860331990027746249921808, 4.54294338165024402937305069733, 4.74623200455836089600652149511, 5.02926382336883831668377067466, 5.40548779991955172021917267478, 5.41011017483418362932083092950, 5.71789386427427522169938518220, 5.94583115840277757570483074124, 6.33353520234343961750241681204, 6.49149435301475766187675589905, 6.65067942891167034305187767781, 6.82774272324287118620274547717, 7.09446953773647332066480738063, 7.24855481526897250176318709932
