| L(s) = 1 | + 4·3-s − 4·5-s + 7-s + 4·9-s − 6·11-s − 16·15-s − 2·17-s − 6·19-s + 4·21-s + 9·23-s − 2·25-s − 9·27-s + 3·29-s − 5·31-s − 24·33-s − 4·35-s − 12·37-s − 3·41-s + 17·43-s − 16·45-s − 8·47-s − 18·49-s − 8·51-s − 15·53-s + 24·55-s − 24·57-s − 17·59-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1.78·5-s + 0.377·7-s + 4/3·9-s − 1.80·11-s − 4.13·15-s − 0.485·17-s − 1.37·19-s + 0.872·21-s + 1.87·23-s − 2/5·25-s − 1.73·27-s + 0.557·29-s − 0.898·31-s − 4.17·33-s − 0.676·35-s − 1.97·37-s − 0.468·41-s + 2.59·43-s − 2.38·45-s − 1.16·47-s − 2.57·49-s − 1.12·51-s − 2.06·53-s + 3.23·55-s − 3.17·57-s − 2.21·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 3 | $A_4\times C_2$ | \( 1 - 4 T + 4 p T^{2} - 23 T^{3} + 4 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 + 4 T + 18 T^{2} + 39 T^{3} + 18 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - T + 19 T^{2} - 13 T^{3} + 19 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 6 T + 24 T^{2} + 91 T^{3} + 24 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 2 T + 8 T^{2} - 59 T^{3} + 8 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 6 T + 48 T^{2} + 201 T^{3} + 48 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 9 T + 68 T^{2} - 301 T^{3} + 68 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 3 T + 27 T^{2} + 77 T^{3} + 27 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 5 T + 71 T^{2} + 213 T^{3} + 71 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 12 T + 96 T^{2} + 511 T^{3} + 96 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 3 T + 98 T^{2} + 275 T^{3} + 98 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 17 T + 209 T^{2} - 1575 T^{3} + 209 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 8 T + 118 T^{2} + 765 T^{3} + 118 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 15 T + 185 T^{2} + 1379 T^{3} + 185 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 17 T + 208 T^{2} + 1629 T^{3} + 208 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 28 T + 400 T^{2} + 3703 T^{3} + 400 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 17 T + 253 T^{2} + 2321 T^{3} + 253 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 7 T + 17 T^{2} + 679 T^{3} + 17 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 15 T + 287 T^{2} + 2273 T^{3} + 287 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 13 T + 221 T^{2} - 1677 T^{3} + 221 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 13 T + 275 T^{2} + 2159 T^{3} + 275 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 19 T + 245 T^{2} + 2081 T^{3} + 245 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 5 T + 227 T^{2} + 999 T^{3} + 227 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.293513866314942604343673319485, −7.82825006299335707677608388980, −7.75676872259688994074386363798, −7.65248048153421197475679721469, −7.50367713458633395835358251681, −7.31148742873779972355535284227, −6.82301239355912450611627974940, −6.35247220897548526800338022813, −6.19221037996177768548061136890, −6.07772790592961745437281393328, −5.42850055108814185821668487465, −5.34606336701190402587675282769, −4.88924969925723247885768441569, −4.54100403544312656462712052297, −4.44295596879588873362438749112, −4.34869624424099025713519019355, −3.49275398985374748745368304991, −3.47607418777367848471754266308, −3.44940667839515811880471296497, −2.82036653028143799702074733526, −2.76642299246072448563490119589, −2.69159604885683841091786692571, −1.92840529313985832910240542225, −1.68813178940736720674017509173, −1.49126640892818510131509266027, 0, 0, 0,
1.49126640892818510131509266027, 1.68813178940736720674017509173, 1.92840529313985832910240542225, 2.69159604885683841091786692571, 2.76642299246072448563490119589, 2.82036653028143799702074733526, 3.44940667839515811880471296497, 3.47607418777367848471754266308, 3.49275398985374748745368304991, 4.34869624424099025713519019355, 4.44295596879588873362438749112, 4.54100403544312656462712052297, 4.88924969925723247885768441569, 5.34606336701190402587675282769, 5.42850055108814185821668487465, 6.07772790592961745437281393328, 6.19221037996177768548061136890, 6.35247220897548526800338022813, 6.82301239355912450611627974940, 7.31148742873779972355535284227, 7.50367713458633395835358251681, 7.65248048153421197475679721469, 7.75676872259688994074386363798, 7.82825006299335707677608388980, 8.293513866314942604343673319485
