| L(s) = 1 | + 15·5-s + 10·7-s − 28·11-s − 78·13-s + 11·17-s + 71·19-s + 25·23-s + 150·25-s − 118·29-s + 107·31-s + 150·35-s − 410·37-s − 592·41-s − 52·43-s + 580·47-s − 225·49-s − 169·53-s − 420·55-s − 234·59-s − 673·61-s − 1.17e3·65-s − 386·67-s − 16·71-s − 892·73-s − 280·77-s − 1.26e3·79-s + 1.81e3·83-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.539·7-s − 0.767·11-s − 1.66·13-s + 0.156·17-s + 0.857·19-s + 0.226·23-s + 6/5·25-s − 0.755·29-s + 0.619·31-s + 0.724·35-s − 1.82·37-s − 2.25·41-s − 0.184·43-s + 1.80·47-s − 0.655·49-s − 0.437·53-s − 1.02·55-s − 0.516·59-s − 1.41·61-s − 2.23·65-s − 0.703·67-s − 0.0267·71-s − 1.43·73-s − 0.414·77-s − 1.79·79-s + 2.40·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 - 10 T + 325 T^{2} - 4052 T^{3} + 325 p^{3} T^{4} - 10 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 28 T + 111 p T^{2} + 66112 T^{3} + 111 p^{4} T^{4} + 28 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 6 p T + 579 p T^{2} + 336036 T^{3} + 579 p^{4} T^{4} + 6 p^{7} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 11 T + 9706 T^{2} + 12229 T^{3} + 9706 p^{3} T^{4} - 11 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 71 T + 19688 T^{2} - 915683 T^{3} + 19688 p^{3} T^{4} - 71 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 25 T + 2176 T^{2} + 755303 T^{3} + 2176 p^{3} T^{4} - 25 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 118 T + 15375 T^{2} + 3162436 T^{3} + 15375 p^{3} T^{4} + 118 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 107 T + 26244 T^{2} - 9258271 T^{3} + 26244 p^{3} T^{4} - 107 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 410 T + 126819 T^{2} + 26481628 T^{3} + 126819 p^{3} T^{4} + 410 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 592 T + 301279 T^{2} + 2091704 p T^{3} + 301279 p^{3} T^{4} + 592 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 52 T + 175069 T^{2} + 917504 T^{3} + 175069 p^{3} T^{4} + 52 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 580 T + 309773 T^{2} - 92211384 T^{3} + 309773 p^{3} T^{4} - 580 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 169 T + 419518 T^{2} + 50902417 T^{3} + 419518 p^{3} T^{4} + 169 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 234 T + 385665 T^{2} + 30035844 T^{3} + 385665 p^{3} T^{4} + 234 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 673 T + 749898 T^{2} + 301087949 T^{3} + 749898 p^{3} T^{4} + 673 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 386 T + 549197 T^{2} + 181189028 T^{3} + 549197 p^{3} T^{4} + 386 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 16 T + 828849 T^{2} + 56595352 T^{3} + 828849 p^{3} T^{4} + 16 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 892 T + 466231 T^{2} + 343918592 T^{3} + 466231 p^{3} T^{4} + 892 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 1263 T + 1118004 T^{2} + 741292675 T^{3} + 1118004 p^{3} T^{4} + 1263 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 1815 T + 2216940 T^{2} - 2103744111 T^{3} + 2216940 p^{3} T^{4} - 1815 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 1800 T + 2584479 T^{2} + 2568723640 T^{3} + 2584479 p^{3} T^{4} + 1800 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 840 T + 1820643 T^{2} + 757404304 T^{3} + 1820643 p^{3} T^{4} + 840 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−8.208768646294313864899213949133, −7.62887942969933492738508105929, −7.59660635877892896521686656692, −7.35532530477157760815161778582, −7.04327383011899013703338028602, −6.74605364762560189612765469180, −6.67178211865716351140886402224, −6.09477637700576352216682958323, −6.00292760722784548888660137569, −5.56796250468403975854103642636, −5.24368182510611503017573392707, −5.16413844015957722574866223102, −5.15485093614693081343756011702, −4.64057736575449188732915616260, −4.31197479802190687803266568397, −4.12442143559081914364562233596, −3.51190452491081330795024012305, −3.11953980668775646627903433556, −3.08587906237498753009657249377, −2.60548160361850216264384015986, −2.29361188719796800862648370226, −2.11890310014957767809132705687, −1.43633632054701429626388012016, −1.40064688918069632295298851044, −1.16274996814723423466890002621, 0, 0, 0,
1.16274996814723423466890002621, 1.40064688918069632295298851044, 1.43633632054701429626388012016, 2.11890310014957767809132705687, 2.29361188719796800862648370226, 2.60548160361850216264384015986, 3.08587906237498753009657249377, 3.11953980668775646627903433556, 3.51190452491081330795024012305, 4.12442143559081914364562233596, 4.31197479802190687803266568397, 4.64057736575449188732915616260, 5.15485093614693081343756011702, 5.16413844015957722574866223102, 5.24368182510611503017573392707, 5.56796250468403975854103642636, 6.00292760722784548888660137569, 6.09477637700576352216682958323, 6.67178211865716351140886402224, 6.74605364762560189612765469180, 7.04327383011899013703338028602, 7.35532530477157760815161778582, 7.59660635877892896521686656692, 7.62887942969933492738508105929, 8.208768646294313864899213949133
