| L(s) = 1 | + 3·2-s − 3·3-s + 6·4-s − 9·6-s + 10·8-s + 6·9-s − 3·11-s − 18·12-s + 15·16-s + 3·17-s + 18·18-s − 9·19-s − 9·22-s + 3·23-s − 30·24-s − 10·27-s + 9·29-s − 6·31-s + 21·32-s + 9·33-s + 9·34-s + 36·36-s + 21·37-s − 27·38-s + 12·41-s + 3·43-s − 18·44-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.73·3-s + 3·4-s − 3.67·6-s + 3.53·8-s + 2·9-s − 0.904·11-s − 5.19·12-s + 15/4·16-s + 0.727·17-s + 4.24·18-s − 2.06·19-s − 1.91·22-s + 0.625·23-s − 6.12·24-s − 1.92·27-s + 1.67·29-s − 1.07·31-s + 3.71·32-s + 1.56·33-s + 1.54·34-s + 6·36-s + 3.45·37-s − 4.37·38-s + 1.87·41-s + 0.457·43-s − 2.71·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.25434632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.25434632\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{3} \) |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $D_{6}$ | \( 1 - 4 p T^{3} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 17 | $S_4\times C_2$ | \( 1 - 3 T - 6 T^{2} + 37 T^{3} - 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 9 T + 69 T^{2} + 314 T^{3} + 69 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 3 T + 42 T^{2} - 49 T^{3} + 42 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 9 T + 39 T^{2} - 6 p T^{3} + 39 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 6 T + 45 T^{2} + 180 T^{3} + 45 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 21 T + 243 T^{2} - 1782 T^{3} + 243 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 12 T + 96 T^{2} - 678 T^{3} + 96 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 3 T - 3 T^{2} + 146 T^{3} - 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 3 T + 114 T^{2} - 273 T^{3} + 114 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{3} \) |
| 59 | $S_4\times C_2$ | \( 1 + 9 T + 129 T^{2} + 1014 T^{3} + 129 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 120 T^{2} + 830 T^{3} + 120 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 6 T + 138 T^{2} - 592 T^{3} + 138 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 9 T + 105 T^{2} - 1170 T^{3} + 105 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 99 T^{2} - 480 T^{3} + 99 p T^{4} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 12 T + 270 T^{2} - 1880 T^{3} + 270 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 18 T + 282 T^{2} - 2824 T^{3} + 282 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 12 T + 255 T^{2} + 2120 T^{3} + 255 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{3} \) |
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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
−7.65638415853180323097962986008, −7.22879310715581657192938050079, −6.86474686234137393001447906670, −6.83537775653755787055218829842, −6.53097533673180727057813588425, −6.18265288791516062719991365611, −6.09549255720244470724635634307, −5.67461840839171908471493341537, −5.66638775860792254168616195747, −5.51603538273905658558150676924, −5.08012026673535512376680628133, −4.68664464047049222632433931200, −4.63538443336979284942494968149, −4.27568216908515599109311923907, −4.16694523396467356403655840179, −4.02297614110335763700332428568, −3.38250537199491794706764140625, −3.14767287423707492032674949862, −2.87659953639650051293254124953, −2.24077992026809123306857026515, −2.17400576853987307769344817120, −2.14266294591869867898616632692, −1.16164945326579604474909161655, −0.74561746532090103070418096808, −0.70023435125508330428256770482,
0.70023435125508330428256770482, 0.74561746532090103070418096808, 1.16164945326579604474909161655, 2.14266294591869867898616632692, 2.17400576853987307769344817120, 2.24077992026809123306857026515, 2.87659953639650051293254124953, 3.14767287423707492032674949862, 3.38250537199491794706764140625, 4.02297614110335763700332428568, 4.16694523396467356403655840179, 4.27568216908515599109311923907, 4.63538443336979284942494968149, 4.68664464047049222632433931200, 5.08012026673535512376680628133, 5.51603538273905658558150676924, 5.66638775860792254168616195747, 5.67461840839171908471493341537, 6.09549255720244470724635634307, 6.18265288791516062719991365611, 6.53097533673180727057813588425, 6.83537775653755787055218829842, 6.86474686234137393001447906670, 7.22879310715581657192938050079, 7.65638415853180323097962986008
