| L(s) = 1 | + 16·23-s + 8·25-s − 8·37-s + 16·47-s + 4·49-s + 32·59-s + 8·61-s + 16·71-s + 16·73-s − 32·83-s + 32·97-s − 32·109-s − 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 36·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 3.33·23-s + 8/5·25-s − 1.31·37-s + 2.33·47-s + 4/7·49-s + 4.16·59-s + 1.02·61-s + 1.89·71-s + 1.87·73-s − 3.51·83-s + 3.24·97-s − 3.06·109-s − 2.54·121-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 − 2.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.897519523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.897519523\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 34 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 706 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 4354 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 1094 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 7330 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3094 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 7522 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 79 | $C_4\times C_2$ | \( 1 - 164 T^{2} + 18054 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 16 T + 222 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 192 T^{2} + 20450 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 226 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.05022328978353077105683881979, −6.89905867450471473372043497959, −6.76024066656569152880103174150, −6.53131637681779322283367766950, −6.07729898028013508789540023631, −5.77471564526972110438194069593, −5.58711389278937596687551122010, −5.42205513459670754041838068018, −5.25378497952992784352540967552, −4.92732105823723651188326058489, −4.87537088238943367927189521620, −4.53532908422761197339466836864, −4.29028695802397280129765324100, −3.87139355916767467356030409684, −3.64912377682849718249442880637, −3.48561489368504187856046285830, −3.36945640649224550823612231383, −2.72677888271569562030243528530, −2.53141991830990794282502796096, −2.46970747226928753047365743098, −2.25738184363746045780969382042, −1.38847680784358031150558796350, −1.33243344650193482672169034751, −0.77942883312104095853371183816, −0.64734792517724004595993496822,
0.64734792517724004595993496822, 0.77942883312104095853371183816, 1.33243344650193482672169034751, 1.38847680784358031150558796350, 2.25738184363746045780969382042, 2.46970747226928753047365743098, 2.53141991830990794282502796096, 2.72677888271569562030243528530, 3.36945640649224550823612231383, 3.48561489368504187856046285830, 3.64912377682849718249442880637, 3.87139355916767467356030409684, 4.29028695802397280129765324100, 4.53532908422761197339466836864, 4.87537088238943367927189521620, 4.92732105823723651188326058489, 5.25378497952992784352540967552, 5.42205513459670754041838068018, 5.58711389278937596687551122010, 5.77471564526972110438194069593, 6.07729898028013508789540023631, 6.53131637681779322283367766950, 6.76024066656569152880103174150, 6.89905867450471473372043497959, 7.05022328978353077105683881979
