| L(s) = 1 | − 2·2-s + 3·4-s − 3·7-s − 4·8-s − 3·11-s − 5·13-s + 6·14-s + 3·16-s − 4·17-s − 19-s + 6·22-s + 10·26-s − 9·28-s − 2·29-s + 17·31-s − 6·32-s + 8·34-s + 2·38-s − 4·41-s − 17·43-s − 9·44-s − 10·47-s − 5·49-s − 15·52-s − 6·53-s + 12·56-s + 4·58-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.13·7-s − 1.41·8-s − 0.904·11-s − 1.38·13-s + 1.60·14-s + 3/4·16-s − 0.970·17-s − 0.229·19-s + 1.27·22-s + 1.96·26-s − 1.70·28-s − 0.371·29-s + 3.05·31-s − 1.06·32-s + 1.37·34-s + 0.324·38-s − 0.624·41-s − 2.59·43-s − 1.35·44-s − 1.45·47-s − 5/7·49-s − 2.08·52-s − 0.824·53-s + 1.60·56-s + 0.525·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $D_{6}$ | \( 1 + p T + T^{2} + p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 3 T + 2 p T^{2} + 25 T^{3} + 2 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 5 T + 3 p T^{2} + 122 T^{3} + 3 p^{2} T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 26 T^{2} + 158 T^{3} + 26 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + T + 2 p T^{2} + 63 T^{3} + 2 p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 26 T^{2} - 58 T^{3} + 26 p T^{4} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 63 T^{2} + 132 T^{3} + 63 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 17 T + 181 T^{2} - 1190 T^{3} + 181 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 36 T^{2} - 34 T^{3} + 36 p T^{4} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 4 T + 122 T^{2} + 326 T^{3} + 122 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 17 T + 97 T^{2} + 362 T^{3} + 97 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 10 T + 166 T^{2} + 948 T^{3} + 166 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 11 T^{2} - 188 T^{3} + 11 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 6 T + 146 T^{2} - 572 T^{3} + 146 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 3 T + 95 T^{2} + 10 p T^{3} + 95 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 7 T + 9 T^{2} - 650 T^{3} + 9 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 26 T + 430 T^{2} + 4272 T^{3} + 430 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 10 T + 175 T^{2} - 988 T^{3} + 175 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 6 T + 200 T^{2} - 736 T^{3} + 200 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 6 T + 33 T^{2} + 4 p T^{3} + 33 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 2 T + 167 T^{2} + 28 T^{3} + 167 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 29 T + 366 T^{2} + 3473 T^{3} + 366 p T^{4} + 29 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.458737548377580695036661566539, −8.104947746330738909092782426955, −7.905886088552100240674021226674, −7.50038649977883579061731863671, −7.25534424039539983996811397076, −7.18531176456174823426168557059, −6.64789392437869995997390514080, −6.55887414126325266496825849823, −6.50066653493361831697712568066, −6.12720421403050291464411139296, −5.89746453641599505905124099967, −5.34610289527388845902917907609, −5.19192964780028953343473014372, −4.69556144045587071048059605826, −4.59842634137726189167702190127, −4.58429675327292667735094426449, −3.73750925151617434511546302203, −3.52002541649683301792237737810, −3.16049985051916385605617533010, −2.92911904526146568130742819605, −2.57707367499413580526006567776, −2.27511628054860443571685882265, −2.05164760275002550134181489637, −1.41780746253486657942500880596, −1.16645455265975697672018915996, 0, 0, 0,
1.16645455265975697672018915996, 1.41780746253486657942500880596, 2.05164760275002550134181489637, 2.27511628054860443571685882265, 2.57707367499413580526006567776, 2.92911904526146568130742819605, 3.16049985051916385605617533010, 3.52002541649683301792237737810, 3.73750925151617434511546302203, 4.58429675327292667735094426449, 4.59842634137726189167702190127, 4.69556144045587071048059605826, 5.19192964780028953343473014372, 5.34610289527388845902917907609, 5.89746453641599505905124099967, 6.12720421403050291464411139296, 6.50066653493361831697712568066, 6.55887414126325266496825849823, 6.64789392437869995997390514080, 7.18531176456174823426168557059, 7.25534424039539983996811397076, 7.50038649977883579061731863671, 7.905886088552100240674021226674, 8.104947746330738909092782426955, 8.458737548377580695036661566539
