| L(s) = 1 | − 3·2-s + 7·4-s + 3·5-s − 14·8-s − 9·10-s + 5·11-s + 21·16-s − 6·17-s − 2·19-s + 21·20-s − 15·22-s − 3·23-s + 6·25-s − 15·29-s − 11·31-s − 28·32-s + 18·34-s + 17·37-s + 6·38-s − 42·40-s + 10·41-s − 26·43-s + 35·44-s + 9·46-s − 10·47-s − 14·49-s − 18·50-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 7/2·4-s + 1.34·5-s − 4.94·8-s − 2.84·10-s + 1.50·11-s + 21/4·16-s − 1.45·17-s − 0.458·19-s + 4.69·20-s − 3.19·22-s − 0.625·23-s + 6/5·25-s − 2.78·29-s − 1.97·31-s − 4.94·32-s + 3.08·34-s + 2.79·37-s + 0.973·38-s − 6.64·40-s + 1.56·41-s − 3.96·43-s + 5.27·44-s + 1.32·46-s − 1.45·47-s − 2·49-s − 2.54·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_6$ | \( 1 + 3 T + p T^{2} - T^{3} + p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 2 p T^{2} + p T^{3} + 2 p^{2} T^{4} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 5 T + 32 T^{2} - 97 T^{3} + 32 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 6 T + 35 T^{2} + 100 T^{3} + 35 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 2 T + 28 T^{2} + 5 T^{3} + 28 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 3 T + p T^{2} - T^{3} + p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_6$ | \( 1 + 15 T + 155 T^{2} + 967 T^{3} + 155 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 11 T + 68 T^{2} + 303 T^{3} + 68 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 17 T + 191 T^{2} - 1371 T^{3} + 191 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 10 T + 112 T^{2} - 807 T^{3} + 112 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 26 T + 338 T^{2} + 2727 T^{3} + 338 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 10 T + 172 T^{2} + 969 T^{3} + 172 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 11 T + 57 T^{2} + 87 T^{3} + 57 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + T + 63 T^{2} - 9 T^{3} + 63 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 22 T + 342 T^{2} - 3061 T^{3} + 342 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 16 T + 200 T^{2} + 1527 T^{3} + 200 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 28 T + 444 T^{2} - 4557 T^{3} + 444 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 4 T + 124 T^{2} - 667 T^{3} + 124 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 6 T + 2 p T^{2} - 1061 T^{3} + 2 p^{2} T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 5 T - 25 T^{2} + 19 T^{3} - 25 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 2 T + 252 T^{2} - 327 T^{3} + 252 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 5 T + 234 T^{2} + 677 T^{3} + 234 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.18961190565451005710431813538, −6.99435425906805003379328774415, −6.94281768928344149055958837701, −6.58072238041545806235753578986, −6.48343026125418053638686984476, −6.33422352700186524468381900865, −6.07009049256669712473649726337, −6.02102610696569499660407652580, −5.47777530410111842797644926547, −5.39931444298427051374072285010, −5.00875372486309072726558444320, −4.69282628753967361673502424330, −4.63978313881497832163514193035, −3.83442200245848607974886249874, −3.78092358056871476818721158232, −3.72862861275838377899154887231, −3.42630288301693866463196223153, −2.69653358862132169579040377132, −2.57290533550262596138727757840, −2.51244125426976013765003863354, −1.93930295517690778426521293765, −1.93271914388934861996803277340, −1.61892923925009960658245018152, −1.23928442521590381211279681186, −1.18610395070062165827795907169, 0, 0, 0,
1.18610395070062165827795907169, 1.23928442521590381211279681186, 1.61892923925009960658245018152, 1.93271914388934861996803277340, 1.93930295517690778426521293765, 2.51244125426976013765003863354, 2.57290533550262596138727757840, 2.69653358862132169579040377132, 3.42630288301693866463196223153, 3.72862861275838377899154887231, 3.78092358056871476818721158232, 3.83442200245848607974886249874, 4.63978313881497832163514193035, 4.69282628753967361673502424330, 5.00875372486309072726558444320, 5.39931444298427051374072285010, 5.47777530410111842797644926547, 6.02102610696569499660407652580, 6.07009049256669712473649726337, 6.33422352700186524468381900865, 6.48343026125418053638686984476, 6.58072238041545806235753578986, 6.94281768928344149055958837701, 6.99435425906805003379328774415, 7.18961190565451005710431813538
