| L(s) = 1 | + 4·7-s + 12·17-s + 28·23-s + 12·25-s + 8·31-s − 12·41-s + 10·49-s + 36·71-s + 24·79-s + 12·89-s − 16·97-s + 8·103-s − 48·113-s + 48·119-s − 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 112·161-s + 163-s + 167-s + 52·169-s + 173-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 2.91·17-s + 5.83·23-s + 12/5·25-s + 1.43·31-s − 1.87·41-s + 10/7·49-s + 4.27·71-s + 2.70·79-s + 1.27·89-s − 1.62·97-s + 0.788·103-s − 4.51·113-s + 4.40·119-s − 1.09·121-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 + 8.82·161-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(19.68803765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(19.68803765\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 74 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 12 T^{2} + 170 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $D_{4}$ | \( ( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 54 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 14 T + 4 p T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 3210 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 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 | $D_4\times C_2$ | \( 1 - 108 T^{2} + 6806 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 212 T^{2} + 18486 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 212 T^{2} + 19446 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 18 T + 196 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 108 T^{2} + 14966 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 6 T + 184 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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
−5.81376226581405424329276779281, −5.69837162815793733743422769822, −5.27241543775451807756315950856, −5.24520283377729367340064051912, −5.23560054577309961820052243269, −4.91929539641235977530414859969, −4.87023707650429559179305347141, −4.86226879844422622990943654673, −4.58567366077884012270527182137, −4.12431909850271278332690868730, −3.93108931708256163490729130901, −3.55624608294188957560913060744, −3.55482998789439348426336826975, −3.18553681017874528894400047915, −3.06297750811738445298997105722, −2.85559887381170073142230464144, −2.80813150945835635471898823782, −2.39894017068598488140019256689, −2.13552997269635036735700175123, −1.74221186388871360461502268628, −1.30734350205996572313445701465, −1.28920741656391630514287370364, −0.860752412614534256915137284789, −0.857754991123173436576595070923, −0.71466605925668100200645110581,
0.71466605925668100200645110581, 0.857754991123173436576595070923, 0.860752412614534256915137284789, 1.28920741656391630514287370364, 1.30734350205996572313445701465, 1.74221186388871360461502268628, 2.13552997269635036735700175123, 2.39894017068598488140019256689, 2.80813150945835635471898823782, 2.85559887381170073142230464144, 3.06297750811738445298997105722, 3.18553681017874528894400047915, 3.55482998789439348426336826975, 3.55624608294188957560913060744, 3.93108931708256163490729130901, 4.12431909850271278332690868730, 4.58567366077884012270527182137, 4.86226879844422622990943654673, 4.87023707650429559179305347141, 4.91929539641235977530414859969, 5.23560054577309961820052243269, 5.24520283377729367340064051912, 5.27241543775451807756315950856, 5.69837162815793733743422769822, 5.81376226581405424329276779281
