| L(s) = 1 | + 4·2-s + 14·4-s + 24·8-s − 6·9-s + 33·16-s − 24·18-s + 12·23-s − 38·25-s − 16·29-s − 8·32-s − 84·36-s + 48·46-s − 2·49-s − 152·50-s − 64·58-s − 78·64-s + 20·71-s − 144·72-s + 21·81-s + 168·92-s − 8·98-s − 532·100-s − 224·116-s + 82·121-s + 127-s − 236·128-s + 131-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 7·4-s + 8.48·8-s − 2·9-s + 33/4·16-s − 5.65·18-s + 2.50·23-s − 7.59·25-s − 2.97·29-s − 1.41·32-s − 14·36-s + 7.07·46-s − 2/7·49-s − 21.4·50-s − 8.40·58-s − 9.75·64-s + 2.37·71-s − 16.9·72-s + 7/3·81-s + 17.5·92-s − 0.808·98-s − 53.1·100-s − 20.7·116-s + 7.45·121-s + 0.0887·127-s − 20.8·128-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.070149832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.070149832\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + T^{2} )^{6} \) |
| 7 | \( 1 + 2 T^{2} + 31 T^{4} + 68 p T^{6} + 31 p^{2} T^{8} + 2 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 - 6 T + 41 T^{2} - 292 T^{3} + 41 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| good | 2 | \( ( 1 - T - T^{2} + p^{2} T^{3} - p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 5 | \( ( 1 + 19 T^{2} + 188 T^{4} + 1152 T^{6} + 188 p^{2} T^{8} + 19 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 11 | \( ( 1 - 41 T^{2} + 882 T^{4} - 11969 T^{6} + 882 p^{2} T^{8} - 41 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( ( 1 - 55 T^{2} + 1360 T^{4} - 21160 T^{6} + 1360 p^{2} T^{8} - 55 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 17 | \( ( 1 + 40 T^{2} + 804 T^{4} + 12250 T^{6} + 804 p^{2} T^{8} + 40 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 + 89 T^{2} + 3682 T^{4} + 89201 T^{6} + 3682 p^{2} T^{8} + 89 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 + 4 T + 68 T^{2} + 182 T^{3} + 68 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 31 | \( ( 1 - 132 T^{2} + 8048 T^{4} - 304754 T^{6} + 8048 p^{2} T^{8} - 132 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 - 108 T^{2} + 4716 T^{4} - 153214 T^{6} + 4716 p^{2} T^{8} - 108 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 41 | \( ( 1 - 57 T^{2} + p^{2} T^{4} )^{6} \) |
| 43 | \( ( 1 - 163 T^{2} + 13676 T^{4} - 731004 T^{6} + 13676 p^{2} T^{8} - 163 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 - 85 T^{2} + 3327 T^{4} - 73030 T^{6} + 3327 p^{2} T^{8} - 85 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 - 232 T^{2} + 25565 T^{4} - 1704261 T^{6} + 25565 p^{2} T^{8} - 232 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 59 | \( ( 1 - 222 T^{2} + 22283 T^{4} - 1494539 T^{6} + 22283 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 + 172 T^{2} + 20417 T^{4} + 1435485 T^{6} + 20417 p^{2} T^{8} + 172 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 67 | \( ( 1 - 303 T^{2} + 43476 T^{4} - 3686564 T^{6} + 43476 p^{2} T^{8} - 303 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 - 5 T + 160 T^{2} - 582 T^{3} + 160 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 73 | \( ( 1 + 12 T^{2} + 8168 T^{4} + 360166 T^{6} + 8168 p^{2} T^{8} + 12 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 - 88 T^{2} - 4604 T^{4} + 891542 T^{6} - 4604 p^{2} T^{8} - 88 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 83 | \( ( 1 + 28 T^{2} + 14036 T^{4} + 184614 T^{6} + 14036 p^{2} T^{8} + 28 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( ( 1 + 363 T^{2} + 62396 T^{4} + 6748280 T^{6} + 62396 p^{2} T^{8} + 363 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 97 | \( ( 1 - 32 T^{2} + 10076 T^{4} - 53046 T^{6} + 10076 p^{2} T^{8} - 32 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.60781188683995643858375371487, −3.41430851992590152471027860783, −3.31548534409300025987463814489, −3.28964389272066275679358297939, −3.11942131376799219835572476234, −3.11429098855797465091356591436, −3.10848759635813834368698478809, −3.08992512950987519187376335928, −3.06622797432229649902727759816, −2.88329640370077421258984030988, −2.47194957969292552127991902834, −2.39689420420267530762264374932, −2.38302198315059345400285523448, −2.14028122094074734697139140559, −2.14013534385528242164906253730, −2.04530008041290297681099442370, −1.89986457315361315310064572148, −1.89767652545717616058405754272, −1.85124753333109692068992102188, −1.85077960904515098745878686299, −1.13918705485763791823732842686, −1.09091186066105895264905118689, −0.69374506625282558242126893483, −0.46040183125985371346753871338, −0.11944497634423176449690490224,
0.11944497634423176449690490224, 0.46040183125985371346753871338, 0.69374506625282558242126893483, 1.09091186066105895264905118689, 1.13918705485763791823732842686, 1.85077960904515098745878686299, 1.85124753333109692068992102188, 1.89767652545717616058405754272, 1.89986457315361315310064572148, 2.04530008041290297681099442370, 2.14013534385528242164906253730, 2.14028122094074734697139140559, 2.38302198315059345400285523448, 2.39689420420267530762264374932, 2.47194957969292552127991902834, 2.88329640370077421258984030988, 3.06622797432229649902727759816, 3.08992512950987519187376335928, 3.10848759635813834368698478809, 3.11429098855797465091356591436, 3.11942131376799219835572476234, 3.28964389272066275679358297939, 3.31548534409300025987463814489, 3.41430851992590152471027860783, 3.60781188683995643858375371487
Plot not available for L-functions of degree greater than 10.
