| L(s) = 1 | + 10·9-s + 12·13-s + 24·25-s + 12·29-s − 12·41-s − 28·73-s + 43·81-s + 120·117-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 10/3·9-s + 3.32·13-s + 24/5·25-s + 2.22·29-s − 1.87·41-s − 3.27·73-s + 43/9·81-s + 11.0·117-s − 3.63·121-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 + 0.769·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.83177438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.83177438\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} )^{2}( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | \( ( 1 - 4 T + p T^{2} )^{4}( 1 + 4 T + p T^{2} )^{4} \) |
| 7 | \( ( 1 - 34 T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | \( ( 1 + 48 T^{2} + 1166 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 61 T^{2} + 2184 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 3 T + 76 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 144 T^{2} + 8750 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 101 T^{2} + 5112 T^{4} - 101 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 144 T^{2} + 10670 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 16 T^{2} - 3186 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 157 T^{2} + 15576 T^{4} - 157 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 12 T^{2} - 6490 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 84 T^{2} + 15494 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 160 T^{2} + 14526 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.98573718291498214879879462522, −3.94917525487236248919501004002, −3.86643229533179116727339986816, −3.59867136976365610507051322700, −3.55508746116791929720821648122, −3.46765579041632211884321230118, −3.27960330709665001130984910252, −3.03071508207904393411986155842, −3.01680760661376665038394101911, −2.85823240932050662247370220485, −2.82025682825506168179550880429, −2.65022720715931748995550061551, −2.63868532254061816881280132492, −2.17185367184374927802414299988, −2.12002115731792375391199616927, −1.73307970077896258287824002077, −1.71139384390780593539390329590, −1.59713655145455717162693490448, −1.52491111323609353423318033514, −1.21052030971962890031199231617, −1.18192244783552880258164006393, −0.911062001053349251374429796405, −0.874974666319179767841936835842, −0.865242153117852734282300504902, −0.22243799873405784320089691747,
0.22243799873405784320089691747, 0.865242153117852734282300504902, 0.874974666319179767841936835842, 0.911062001053349251374429796405, 1.18192244783552880258164006393, 1.21052030971962890031199231617, 1.52491111323609353423318033514, 1.59713655145455717162693490448, 1.71139384390780593539390329590, 1.73307970077896258287824002077, 2.12002115731792375391199616927, 2.17185367184374927802414299988, 2.63868532254061816881280132492, 2.65022720715931748995550061551, 2.82025682825506168179550880429, 2.85823240932050662247370220485, 3.01680760661376665038394101911, 3.03071508207904393411986155842, 3.27960330709665001130984910252, 3.46765579041632211884321230118, 3.55508746116791929720821648122, 3.59867136976365610507051322700, 3.86643229533179116727339986816, 3.94917525487236248919501004002, 3.98573718291498214879879462522
Plot not available for L-functions of degree greater than 10.
