| L(s) = 1 | + 12·7-s − 12·23-s − 2·25-s − 24·31-s − 24·41-s + 12·47-s + 68·49-s − 24·71-s − 16·73-s − 48·79-s + 24·89-s + 16·97-s + 12·103-s + 24·113-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 144·161-s + 163-s + 167-s + 28·169-s + 173-s + ⋯ |
| L(s) = 1 | + 4.53·7-s − 2.50·23-s − 2/5·25-s − 4.31·31-s − 3.74·41-s + 1.75·47-s + 68/7·49-s − 2.84·71-s − 1.87·73-s − 5.40·79-s + 2.54·89-s + 1.62·97-s + 1.18·103-s + 2.25·113-s + 1.81·121-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 − 11.3·161-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3933214378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3933214378\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $D_{4}$ | \( ( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2154 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 1974 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 13002 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 - 308 T^{2} + 37386 T^{4} - 308 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.09967931245248607121931903309, −5.85442489273614801452042765470, −5.77509651081506931069624856806, −5.67542120473860773913815470587, −5.31710967860216297578110082162, −5.11473219255748155624993295499, −4.97020720747110824894902887856, −4.90440615162653628871812081313, −4.43817955702520722646538019890, −4.41992202938802483792358204239, −4.38415104965173948361661639438, −3.88200187039545453162701796307, −3.79785743374395668875625244040, −3.60385447524775573203031901879, −3.33549009450246415815202602284, −2.91253215631202599062806034442, −2.69053043363986135367585423118, −2.23034202897717829584604972436, −2.02130523769817301593823351542, −1.70279211486592211195792567403, −1.67537820467108124592958741136, −1.61433707470542301626377023791, −1.48743359418671449145646546010, −0.76104772233906374179756945826, −0.080985514010113818105498405207,
0.080985514010113818105498405207, 0.76104772233906374179756945826, 1.48743359418671449145646546010, 1.61433707470542301626377023791, 1.67537820467108124592958741136, 1.70279211486592211195792567403, 2.02130523769817301593823351542, 2.23034202897717829584604972436, 2.69053043363986135367585423118, 2.91253215631202599062806034442, 3.33549009450246415815202602284, 3.60385447524775573203031901879, 3.79785743374395668875625244040, 3.88200187039545453162701796307, 4.38415104965173948361661639438, 4.41992202938802483792358204239, 4.43817955702520722646538019890, 4.90440615162653628871812081313, 4.97020720747110824894902887856, 5.11473219255748155624993295499, 5.31710967860216297578110082162, 5.67542120473860773913815470587, 5.77509651081506931069624856806, 5.85442489273614801452042765470, 6.09967931245248607121931903309
