| L(s) = 1 | − 6·11-s − 8·19-s + 12·29-s − 16·31-s − 6·41-s + 5·49-s − 18·59-s − 10·61-s + 36·71-s − 20·79-s + 48·89-s + 6·101-s − 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 17·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.80·11-s − 1.83·19-s + 2.22·29-s − 2.87·31-s − 0.937·41-s + 5/7·49-s − 2.34·59-s − 1.28·61-s + 4.27·71-s − 2.25·79-s + 5.08·89-s + 0.597·101-s − 0.191·109-s + 1/11·121-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 + 1.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7393317030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7393317030\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 5 T^{2} + 93 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 17 T^{2} + 129 T^{4} - 17 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 5 T^{2} - 327 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 - 65 T^{2} + 2013 T^{4} - 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 113 T^{2} + 5649 T^{4} - 113 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 3 T - 17 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 137 T^{2} + 8109 T^{4} - 137 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 185 T^{2} + 14073 T^{4} - 185 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 9 T + 127 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 5 T + 117 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - 170 T^{2} + 15003 T^{4} - 170 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 121 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−5.41387041403962100526961760154, −5.31491082978398197103130200962, −5.13358877322944366909728541985, −4.83609047525579764375611006993, −4.71283706551914792769701944958, −4.68377260700630546865821227095, −4.66939831252681329352168391057, −4.08199105041108643945922695173, −3.86437782856170392515247448501, −3.75230822145319863106847380486, −3.73574931834477318907851657219, −3.50263357319905574674097347266, −3.05720642065196295874316943227, −2.95482109991609222672881014180, −2.91083784088174534269873659026, −2.33019525155835964165707543102, −2.27996356424180683821930570853, −2.24077725466411603929953345379, −2.16157800486398701998417842853, −1.61542009027964851319180446561, −1.32576602300885447917966658340, −1.26392896710841313195533349435, −0.839215439299761580986296170765, −0.26960896922984068679938693841, −0.21202165684284673663922051451,
0.21202165684284673663922051451, 0.26960896922984068679938693841, 0.839215439299761580986296170765, 1.26392896710841313195533349435, 1.32576602300885447917966658340, 1.61542009027964851319180446561, 2.16157800486398701998417842853, 2.24077725466411603929953345379, 2.27996356424180683821930570853, 2.33019525155835964165707543102, 2.91083784088174534269873659026, 2.95482109991609222672881014180, 3.05720642065196295874316943227, 3.50263357319905574674097347266, 3.73574931834477318907851657219, 3.75230822145319863106847380486, 3.86437782856170392515247448501, 4.08199105041108643945922695173, 4.66939831252681329352168391057, 4.68377260700630546865821227095, 4.71283706551914792769701944958, 4.83609047525579764375611006993, 5.13358877322944366909728541985, 5.31491082978398197103130200962, 5.41387041403962100526961760154
