| L(s) = 1 | + 5·3-s − 2·5-s − 35·7-s − 4·9-s + 28·11-s + 109·13-s − 10·15-s − 123·17-s − 57·19-s − 175·21-s − 193·23-s − 92·25-s + 133·27-s + 297·29-s − 140·31-s + 140·33-s + 70·35-s − 38·37-s + 545·39-s + 736·41-s + 514·43-s + 8·45-s + 134·47-s + 77·49-s − 615·51-s − 311·53-s − 56·55-s + ⋯ |
| L(s) = 1 | + 0.962·3-s − 0.178·5-s − 1.88·7-s − 0.148·9-s + 0.767·11-s + 2.32·13-s − 0.172·15-s − 1.75·17-s − 0.688·19-s − 1.81·21-s − 1.74·23-s − 0.735·25-s + 0.947·27-s + 1.90·29-s − 0.811·31-s + 0.738·33-s + 0.338·35-s − 0.168·37-s + 2.23·39-s + 2.80·41-s + 1.82·43-s + 0.0265·45-s + 0.415·47-s + 0.224·49-s − 1.68·51-s − 0.806·53-s − 0.137·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.340085468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.340085468\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 5 T + 29 T^{2} - 298 T^{3} + 29 p^{3} T^{4} - 5 p^{6} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 2 T + 96 T^{2} - 1016 T^{3} + 96 p^{3} T^{4} + 2 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 5 p T + 164 p T^{2} + 23983 T^{3} + 164 p^{4} T^{4} + 5 p^{7} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 28 T + 4182 T^{2} - 74894 T^{3} + 4182 p^{3} T^{4} - 28 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 109 T + 6999 T^{2} - 365490 T^{3} + 6999 p^{3} T^{4} - 109 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 123 T + 17610 T^{2} + 1215235 T^{3} + 17610 p^{3} T^{4} + 123 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 193 T + 39933 T^{2} + 4449614 T^{3} + 39933 p^{3} T^{4} + 193 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 297 T + 71131 T^{2} - 13319430 T^{3} + 71131 p^{3} T^{4} - 297 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 140 T + 62189 T^{2} + 4673384 T^{3} + 62189 p^{3} T^{4} + 140 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 38 T + 100099 T^{2} + 7279012 T^{3} + 100099 p^{3} T^{4} + 38 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 736 T + 354551 T^{2} - 108718336 T^{3} + 354551 p^{3} T^{4} - 736 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 514 T + 224130 T^{2} - 56635248 T^{3} + 224130 p^{3} T^{4} - 514 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 134 T + 8542 T^{2} + 31064212 T^{3} + 8542 p^{3} T^{4} - 134 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 311 T + 285651 T^{2} + 106221286 T^{3} + 285651 p^{3} T^{4} + 311 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 199 T + 35145 T^{2} - 35868602 T^{3} + 35145 p^{3} T^{4} + 199 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 56 T + 192812 T^{2} - 95078710 T^{3} + 192812 p^{3} T^{4} + 56 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 509 T + 553245 T^{2} - 128354670 T^{3} + 553245 p^{3} T^{4} - 509 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 874 T + 862737 T^{2} + 513398212 T^{3} + 862737 p^{3} T^{4} + 874 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 203 T + 1104798 T^{2} - 157466799 T^{3} + 1104798 p^{3} T^{4} - 203 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 242 T + 502613 T^{2} - 37031420 T^{3} + 502613 p^{3} T^{4} - 242 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 62 T + 1258913 T^{2} - 154550580 T^{3} + 1258913 p^{3} T^{4} - 62 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 1764 T + 3042123 T^{2} - 2614453128 T^{3} + 3042123 p^{3} T^{4} - 1764 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 2178 T + 3988819 T^{2} + 4075506692 T^{3} + 3988819 p^{3} T^{4} + 2178 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−8.526197804049030935159546727949, −7.984750399905913164906196604059, −7.75839719741997873403652082637, −7.63769198565172058046091780666, −7.15764438332254677047767978599, −6.63552508447276045883491333895, −6.39027742269832169531713958132, −6.34550811479308048082527021508, −6.24735860126910892257881385268, −6.04492715214467868858643370953, −5.48594760382382128646056107081, −5.18378222670072761907491486471, −4.34327365040136608823990253479, −4.33957174762136568794719438689, −4.15590425709941768154672821537, −3.90058936334656375051940750165, −3.29134913766045925685773541267, −3.13703709621289075040135003987, −3.02765136271079751466516005713, −2.33147895843441372681180368283, −2.16431894413530533290774029349, −1.80182981627755068622544222603, −0.948421154978596482011943523896, −0.855572303946344087239671871350, −0.24295555208557975272126401291,
0.24295555208557975272126401291, 0.855572303946344087239671871350, 0.948421154978596482011943523896, 1.80182981627755068622544222603, 2.16431894413530533290774029349, 2.33147895843441372681180368283, 3.02765136271079751466516005713, 3.13703709621289075040135003987, 3.29134913766045925685773541267, 3.90058936334656375051940750165, 4.15590425709941768154672821537, 4.33957174762136568794719438689, 4.34327365040136608823990253479, 5.18378222670072761907491486471, 5.48594760382382128646056107081, 6.04492715214467868858643370953, 6.24735860126910892257881385268, 6.34550811479308048082527021508, 6.39027742269832169531713958132, 6.63552508447276045883491333895, 7.15764438332254677047767978599, 7.63769198565172058046091780666, 7.75839719741997873403652082637, 7.984750399905913164906196604059, 8.526197804049030935159546727949
