| L(s) = 1 | − 6·3-s + 4·4-s + 21·9-s − 24·12-s + 4·16-s + 14·25-s − 54·27-s + 12·31-s + 84·36-s + 12·37-s − 24·48-s + 24·49-s − 16·64-s − 12·67-s − 84·75-s + 108·81-s − 72·93-s + 52·97-s + 56·100-s − 8·103-s − 216·108-s − 72·111-s + 48·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 2·4-s + 7·9-s − 6.92·12-s + 16-s + 14/5·25-s − 10.3·27-s + 2.15·31-s + 14·36-s + 1.97·37-s − 3.46·48-s + 24/7·49-s − 2·64-s − 1.46·67-s − 9.69·75-s + 12·81-s − 7.46·93-s + 5.27·97-s + 28/5·100-s − 0.788·103-s − 20.7·108-s − 6.83·111-s + 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.289231966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.289231966\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 12 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 29 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 115 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 139 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 60 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 103 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{4} \) |
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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
−7.986697497361886462917083167193, −7.77722250945110279938339856651, −7.75812445158891899318126234406, −7.18697739306360444807803929103, −7.03712886710048223876361270495, −6.92993200743366084091226455429, −6.78230091097659201488024199091, −6.50614445113869667953432807538, −6.15222841573620387389931692050, −6.00548994403929443258488674530, −5.95011197734945801151294373037, −5.55243056192535114148784433983, −5.37884359325909885357014017414, −4.70227713302943475500387310907, −4.68223736606120856919475175536, −4.52495409239622281629128275727, −4.44678010539278222636267633772, −3.68236351084795147659661586266, −3.32151746501759220364165981026, −2.75669808476478408006534926187, −2.47350657375144483036228481245, −2.23232437253991512973222353873, −1.45613867602936343880513462482, −0.983485272520969413425119628521, −0.76114977411735247770952218629,
0.76114977411735247770952218629, 0.983485272520969413425119628521, 1.45613867602936343880513462482, 2.23232437253991512973222353873, 2.47350657375144483036228481245, 2.75669808476478408006534926187, 3.32151746501759220364165981026, 3.68236351084795147659661586266, 4.44678010539278222636267633772, 4.52495409239622281629128275727, 4.68223736606120856919475175536, 4.70227713302943475500387310907, 5.37884359325909885357014017414, 5.55243056192535114148784433983, 5.95011197734945801151294373037, 6.00548994403929443258488674530, 6.15222841573620387389931692050, 6.50614445113869667953432807538, 6.78230091097659201488024199091, 6.92993200743366084091226455429, 7.03712886710048223876361270495, 7.18697739306360444807803929103, 7.75812445158891899318126234406, 7.77722250945110279938339856651, 7.986697497361886462917083167193
