| L(s) = 1 | − 2·2-s + 3-s + 2·4-s + 3·5-s − 2·6-s + 7-s − 5·8-s − 6·10-s + 9·11-s + 2·12-s − 7·13-s − 2·14-s + 3·15-s + 5·16-s + 9·17-s + 8·19-s + 6·20-s + 21-s − 18·22-s − 4·23-s − 5·24-s + 10·25-s + 14·26-s + 2·28-s − 7·29-s − 6·30-s − 3·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s + 1.34·5-s − 0.816·6-s + 0.377·7-s − 1.76·8-s − 1.89·10-s + 2.71·11-s + 0.577·12-s − 1.94·13-s − 0.534·14-s + 0.774·15-s + 5/4·16-s + 2.18·17-s + 1.83·19-s + 1.34·20-s + 0.218·21-s − 3.83·22-s − 0.834·23-s − 1.02·24-s + 2·25-s + 2.74·26-s + 0.377·28-s − 1.29·29-s − 1.09·30-s − 0.538·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.383872450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.383872450\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 7 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 - 9 T + 41 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_2^2:C_4$ | \( 1 + p T + p T^{2} + 5 T^{3} + 11 T^{4} + 5 p T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 - 3 T - T^{2} + 3 T^{3} + 16 T^{4} + 3 p T^{5} - p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 7 T + 6 T^{2} - 49 T^{3} - 181 T^{4} - 49 p T^{5} + 6 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 9 T + 44 T^{2} - 243 T^{3} + 1279 T^{4} - 243 p T^{5} + 44 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 8 T + 45 T^{2} - 268 T^{3} + 1529 T^{4} - 268 p T^{5} + 45 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 7 T - 10 T^{2} - 113 T^{3} + 59 T^{4} - 113 p T^{5} - 10 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 + 3 T - 22 T^{2} - 159 T^{3} + 205 T^{4} - 159 p T^{5} - 22 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 + 6 T - T^{2} - 228 T^{3} - 1331 T^{4} - 228 p T^{5} - p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + T + 35 T^{2} + 99 T^{3} + 1904 T^{4} + 99 p T^{5} + 35 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 14 T + 115 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 16 T + 59 T^{2} + 738 T^{3} - 9721 T^{4} + 738 p T^{5} + 59 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 25 T + 257 T^{2} - 1635 T^{3} + 10244 T^{4} - 1635 p T^{5} + 257 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - T - 43 T^{2} + 347 T^{3} + 2540 T^{4} + 347 p T^{5} - 43 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 19 T + 180 T^{2} - 1781 T^{3} + 17099 T^{4} - 1781 p T^{5} + 180 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 22 T + 250 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 + T + 70 T^{2} + 119 T^{3} + 5709 T^{4} + 119 p T^{5} + 70 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 11 T + 3 T^{2} - 815 T^{3} - 8464 T^{4} - 815 p T^{5} + 3 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 13 T + 15 T^{2} - 293 T^{3} + 8624 T^{4} - 293 p T^{5} + 15 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 14 T - 7 T^{2} - 640 T^{3} - 2059 T^{4} - 640 p T^{5} - 7 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 13 T + 159 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 3 T + 182 T^{2} - 345 T^{3} + 16591 T^{4} - 345 p T^{5} + 182 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.167679652138246790770490067555, −8.428063946224317755795563440833, −8.383139908639715914282471479415, −8.369403849185161650183502390131, −8.333594927742506886912413292225, −7.48651650974909689029648566979, −7.28940874036668505933297838907, −7.03448817445642150582966527089, −6.79245754156162379511386081365, −6.68999949061032228405522961418, −6.57425292831900081127331056514, −5.65999297412013989086556398345, −5.56624524916865205402574191144, −5.46051370661874869784941089377, −5.32247828989768773616474982148, −4.89309863858823884918400045961, −4.02996426462646267312076079424, −3.90439555524014659031966272707, −3.69132467307026662966969430440, −3.00778686719332393471852774768, −2.92988363439295471638258207660, −2.16036572573081923361759851181, −2.09350870517998568886041453917, −1.34198471512577951866034080089, −0.921556623080204601596263201403,
0.921556623080204601596263201403, 1.34198471512577951866034080089, 2.09350870517998568886041453917, 2.16036572573081923361759851181, 2.92988363439295471638258207660, 3.00778686719332393471852774768, 3.69132467307026662966969430440, 3.90439555524014659031966272707, 4.02996426462646267312076079424, 4.89309863858823884918400045961, 5.32247828989768773616474982148, 5.46051370661874869784941089377, 5.56624524916865205402574191144, 5.65999297412013989086556398345, 6.57425292831900081127331056514, 6.68999949061032228405522961418, 6.79245754156162379511386081365, 7.03448817445642150582966527089, 7.28940874036668505933297838907, 7.48651650974909689029648566979, 8.333594927742506886912413292225, 8.369403849185161650183502390131, 8.383139908639715914282471479415, 8.428063946224317755795563440833, 9.167679652138246790770490067555
