| L(s) = 1 | − 32·61-s + 32·81-s + 96·101-s − 128·121-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 | − 4.09·61-s + 32/9·81-s + 9.55·101-s − 11.6·121-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 + 0.0760·173-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 + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{32} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{32} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2298923451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2298923451\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( ( 1 - 20 T^{2} + 438 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| good | 3 | \( ( 1 - 16 T^{4} + 130 T^{8} - 16 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 7 | \( ( 1 + 4 T^{4} + 3270 T^{8} + 4 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{8} \) |
| 13 | \( ( 1 - 464 T^{4} + 108546 T^{8} - 464 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 17 | \( ( 1 - 188 T^{4} + 22278 T^{8} - 188 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 23 | \( ( 1 - 1724 T^{4} + 1264326 T^{8} - 1724 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 29 | \( ( 1 + 20 T^{2} + 1398 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 + 20 T^{2} + 1158 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 37 | \( ( 1 - 4496 T^{4} + 8794050 T^{8} - 4496 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 41 | \( ( 1 - 20 T^{2} - 1242 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 43 | \( ( 1 - 668 T^{4} - 1690842 T^{8} - 668 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 47 | \( ( 1 + 4804 T^{4} + 12946950 T^{8} + 4804 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 - 4304 T^{4} + 8985666 T^{8} - 4304 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 59 | \( ( 1 + 92 T^{2} + 8214 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 61 | \( ( 1 + 4 T + 72 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{8} \) |
| 67 | \( ( 1 - 7184 T^{4} + 43679106 T^{8} - 7184 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 71 | \( ( 1 - 140 T^{2} + 10278 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 - 4028 T^{4} - 587322 T^{8} - 4028 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 + 172 T^{2} + 15174 T^{4} + 172 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 83 | \( ( 1 - 1436 T^{4} + 28338150 T^{8} - 1436 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 + 260 T^{2} + 32358 T^{4} + 260 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 97 | \( ( 1 - 9584 T^{4} + 21152226 T^{8} - 9584 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.32936656157429676419461443927, −2.30593171978247960467183254878, −2.10604419313420433155918270336, −1.97908097687730032188897145907, −1.95911235214491340601538738872, −1.94573386090091691086517695154, −1.83808298059408003824410825166, −1.79426730698468388421505484947, −1.67184817160130096176639649157, −1.65976919461724403039862028708, −1.64885260944374239301955283766, −1.56008981284264509733129550111, −1.39491565782994089831383630474, −1.15264256493763482744146732662, −1.12895041515698391624551416654, −1.11963041425962198252571132837, −0.977808536317234148895461395645, −0.919395067247455861177000634146, −0.882604794986435938474625663670, −0.73493304091693700922122158273, −0.62200325344866657122162414351, −0.60413314744705448782597856402, −0.17145753938675597170757850983, −0.11436679933777393495173407920, −0.07000386117310058156575972127,
0.07000386117310058156575972127, 0.11436679933777393495173407920, 0.17145753938675597170757850983, 0.60413314744705448782597856402, 0.62200325344866657122162414351, 0.73493304091693700922122158273, 0.882604794986435938474625663670, 0.919395067247455861177000634146, 0.977808536317234148895461395645, 1.11963041425962198252571132837, 1.12895041515698391624551416654, 1.15264256493763482744146732662, 1.39491565782994089831383630474, 1.56008981284264509733129550111, 1.64885260944374239301955283766, 1.65976919461724403039862028708, 1.67184817160130096176639649157, 1.79426730698468388421505484947, 1.83808298059408003824410825166, 1.94573386090091691086517695154, 1.95911235214491340601538738872, 1.97908097687730032188897145907, 2.10604419313420433155918270336, 2.30593171978247960467183254878, 2.32936656157429676419461443927
Plot not available for L-functions of degree greater than 10.
