| L(s) = 1 | + 4·9-s − 8·17-s − 48·41-s + 4·49-s + 56·89-s + 16·97-s + 32·113-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 32·153-s + 157-s + 163-s + 167-s − 64·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 4/3·9-s − 1.94·17-s − 7.49·41-s + 4/7·49-s + 5.93·89-s + 1.62·97-s + 3.01·113-s − 4·121-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 − 2.58·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4.92·169-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 + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.200448118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.200448118\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( ( 1 + T )^{8} \) |
| good | 3 | \( ( 1 - 2 T^{2} + 2 p T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 - 2 T^{4} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 2 T^{2} - 18 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 2 p T^{2} + 246 T^{4} + 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 32 T^{2} + 542 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 20 T^{2} + 614 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 70 T^{2} + 2270 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 96 T^{2} + 3934 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 6 T^{2} + 1918 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 16 T^{2} + 2334 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 6 T + p T^{2} )^{8} \) |
| 43 | \( ( 1 + p T^{2} )^{8} \) |
| 47 | \( ( 1 - 76 T^{2} + 3990 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 36 T^{2} + 5110 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 64 T^{2} + 4254 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 2 p T^{2} + 13550 T^{4} + 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 198 T^{2} + 22270 T^{4} + 198 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 108 T^{2} + 13366 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 14 T + 214 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.41928953285448899399222744360, −3.37926087971523589467831646437, −3.23228586651049829946230017548, −3.21185603846594157794415211819, −2.84647320608418880431618570952, −2.81888311706837899187300742557, −2.81775617204067413053823334089, −2.81612591920773637757732566832, −2.48792539798388730236529373617, −2.18397237072354531104334142827, −2.17552766900527159208267616978, −2.07372168691640597569569590620, −2.06857313696572801272399515969, −1.81506666102285907145300543151, −1.80423361459486657065440863991, −1.79276458113586186529590895834, −1.47154987467841227128521499343, −1.38638637356528418544662276461, −1.35045978590092848894307925507, −1.02128377449290384900208719629, −0.77934998324239297417980823976, −0.70419243964762218094621029775, −0.57802400843684658867783097007, −0.22549653087683259657996709590, −0.15717216124819489558650268447,
0.15717216124819489558650268447, 0.22549653087683259657996709590, 0.57802400843684658867783097007, 0.70419243964762218094621029775, 0.77934998324239297417980823976, 1.02128377449290384900208719629, 1.35045978590092848894307925507, 1.38638637356528418544662276461, 1.47154987467841227128521499343, 1.79276458113586186529590895834, 1.80423361459486657065440863991, 1.81506666102285907145300543151, 2.06857313696572801272399515969, 2.07372168691640597569569590620, 2.17552766900527159208267616978, 2.18397237072354531104334142827, 2.48792539798388730236529373617, 2.81612591920773637757732566832, 2.81775617204067413053823334089, 2.81888311706837899187300742557, 2.84647320608418880431618570952, 3.21185603846594157794415211819, 3.23228586651049829946230017548, 3.37926087971523589467831646437, 3.41928953285448899399222744360
Plot not available for L-functions of degree greater than 10.
