| L(s) = 1 | − 4·11-s + 32·29-s + 8·31-s − 32·41-s + 32·59-s + 16·79-s + 24·89-s + 8·101-s + 40·109-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 24·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 5.94·29-s + 1.43·31-s − 4.99·41-s + 4.16·59-s + 1.80·79-s + 2.54·89-s + 0.796·101-s + 3.83·109-s + 0.909·121-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 + 0.0783·163-s + 0.0773·167-s + 1.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.727756681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.727756681\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $D_4\times C_2$ | \( 1 - 24 T^{2} + 430 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 24 T^{2} + 254 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T^{2} - 554 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 43 | $D_4\times C_2$ | \( 1 - 48 T^{2} + 1726 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 5030 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 248 T^{2} + 25566 T^{4} - 248 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 208 T^{2} + 22046 T^{4} - 208 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 + 36 T^{2} + 15814 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.30653848973342163430984386279, −4.99399301375601032310390408506, −4.90197534016633377082783896888, −4.85778947923289633455059963362, −4.73394836238265380228737385388, −4.62699478437363553479218241671, −4.41896528847232058436151533059, −4.16084515173172335263575766950, −3.82182236048240985832930225014, −3.53196375322916491001830512011, −3.51059638381379685621736191110, −3.27835679722124433688943609983, −3.20421711489209350752054282103, −2.84471653252072783398990036833, −2.73324191888146563582573694480, −2.55471312536737335486824210736, −2.30649011110878956994290118102, −2.06397241797772370928374304152, −1.81829356700256586592698255610, −1.81776465080475791567350912836, −1.20284702301033936445697917392, −0.876920776062417562883858176260, −0.824774634586113450034812111642, −0.71916706054640485543823445284, −0.27423337976123978858395533059,
0.27423337976123978858395533059, 0.71916706054640485543823445284, 0.824774634586113450034812111642, 0.876920776062417562883858176260, 1.20284702301033936445697917392, 1.81776465080475791567350912836, 1.81829356700256586592698255610, 2.06397241797772370928374304152, 2.30649011110878956994290118102, 2.55471312536737335486824210736, 2.73324191888146563582573694480, 2.84471653252072783398990036833, 3.20421711489209350752054282103, 3.27835679722124433688943609983, 3.51059638381379685621736191110, 3.53196375322916491001830512011, 3.82182236048240985832930225014, 4.16084515173172335263575766950, 4.41896528847232058436151533059, 4.62699478437363553479218241671, 4.73394836238265380228737385388, 4.85778947923289633455059963362, 4.90197534016633377082783896888, 4.99399301375601032310390408506, 5.30653848973342163430984386279
