| L(s) = 1 | + 8·53-s + 4·81-s + 16·113-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | + 8·53-s + 4·81-s + 16·113-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 5^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 5^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.754260239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.754260239\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T^{8} + T^{16} \) |
| 7 | \( 1 \) |
| good | 3 | \( ( 1 - T^{4} + T^{8} )^{4} \) |
| 11 | \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \) |
| 13 | \( ( 1 + T^{8} )^{4} \) |
| 17 | \( ( 1 - T^{8} + T^{16} )^{2} \) |
| 19 | \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \) |
| 23 | \( ( 1 - T^{4} + T^{8} )^{4} \) |
| 29 | \( ( 1 + T^{4} )^{8} \) |
| 31 | \( ( 1 - T^{2} + T^{4} )^{8} \) |
| 37 | \( ( 1 - T^{4} + T^{8} )^{4} \) |
| 41 | \( ( 1 + T^{8} )^{4} \) |
| 43 | \( ( 1 + T^{4} )^{8} \) |
| 47 | \( ( 1 - T^{4} + T^{8} )^{4} \) |
| 53 | \( ( 1 - T + T^{2} )^{8}( 1 - T^{2} + T^{4} )^{4} \) |
| 59 | \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \) |
| 61 | \( ( 1 - T^{8} + T^{16} )^{2} \) |
| 67 | \( ( 1 - T^{4} + T^{8} )^{4} \) |
| 71 | \( ( 1 - T )^{16}( 1 + T )^{16} \) |
| 73 | \( ( 1 - T^{8} + T^{16} )^{2} \) |
| 79 | \( ( 1 - T^{2} + T^{4} )^{8} \) |
| 83 | \( ( 1 + T^{4} )^{8} \) |
| 89 | \( ( 1 - T^{8} + T^{16} )^{2} \) |
| 97 | \( ( 1 + T^{8} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.28029721270195856769038844325, −2.22473271480746796021135779839, −2.21552425600660151434293592183, −2.19674684513873403396136706128, −2.11563345474380374881956938979, −2.11460060210026892770636273291, −1.90464267110288267970900851895, −1.85631519466179718673316781499, −1.80036553905066311931658170668, −1.65924969479245898213628193011, −1.41574069744748303335841413108, −1.39682455406892899880959016098, −1.35239884219236439702332284446, −1.34256831903997319395807278827, −1.33863972696534547003139124745, −1.28301217238724581554223030146, −1.23375566306066413544447781518, −1.04984633787087071444886035687, −0.869743149276443668558106479813, −0.822077296150696401396145358258, −0.76041929258738686985282340500, −0.68917433027391272190956518837, −0.68871297106483726106070601002, −0.51374411589060138566693037368, −0.21094263446537228234223033387,
0.21094263446537228234223033387, 0.51374411589060138566693037368, 0.68871297106483726106070601002, 0.68917433027391272190956518837, 0.76041929258738686985282340500, 0.822077296150696401396145358258, 0.869743149276443668558106479813, 1.04984633787087071444886035687, 1.23375566306066413544447781518, 1.28301217238724581554223030146, 1.33863972696534547003139124745, 1.34256831903997319395807278827, 1.35239884219236439702332284446, 1.39682455406892899880959016098, 1.41574069744748303335841413108, 1.65924969479245898213628193011, 1.80036553905066311931658170668, 1.85631519466179718673316781499, 1.90464267110288267970900851895, 2.11460060210026892770636273291, 2.11563345474380374881956938979, 2.19674684513873403396136706128, 2.21552425600660151434293592183, 2.22473271480746796021135779839, 2.28029721270195856769038844325
Plot not available for L-functions of degree greater than 10.
