L(s) = 1 | + 2.51e3·3-s + 1.63e4·4-s + 1.00e5·5-s + 1.54e6·9-s − 1.94e7·11-s + 4.12e7·12-s + 2.53e8·15-s + 2.68e8·16-s + 1.65e9·20-s + 1.55e9·23-s + 4.05e9·25-s − 8.15e9·27-s − 5.34e10·31-s − 4.90e10·33-s + 2.52e10·36-s + 1.26e11·37-s − 3.19e11·44-s + 1.55e11·45-s + 7.15e11·47-s + 6.75e11·48-s + 6.78e11·49-s − 2.21e12·53-s − 1.96e12·55-s − 4.95e12·59-s + 4.15e12·60-s + 4.39e12·64-s − 1.16e13·67-s + ⋯ |
L(s) = 1 | + 1.14·3-s + 4-s + 1.29·5-s + 0.322·9-s − 11-s + 1.14·12-s + 1.48·15-s + 16-s + 1.29·20-s + 0.457·23-s + 0.664·25-s − 0.779·27-s − 1.94·31-s − 1.14·33-s + 0.322·36-s + 1.33·37-s − 44-s + 0.416·45-s + 1.41·47-s + 1.14·48-s + 49-s − 1.88·53-s − 1.29·55-s − 1.99·59-s + 1.48·60-s + 64-s − 1.92·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{15}{2})\) |
\(\approx\) |
\(3.651430639\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.651430639\) |
\(L(8)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + p^{7} T \) |
good | 2 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 3 | \( 1 - 2515 T + p^{14} T^{2} \) |
| 5 | \( 1 - 100799 T + p^{14} T^{2} \) |
| 7 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 13 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 17 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 19 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 23 | \( 1 - 1556561195 T + p^{14} T^{2} \) |
| 29 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 31 | \( 1 + 53499153997 T + p^{14} T^{2} \) |
| 37 | \( 1 - 126509871575 T + p^{14} T^{2} \) |
| 41 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 43 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 47 | \( 1 - 715778884850 T + p^{14} T^{2} \) |
| 53 | \( 1 + 2217378708790 T + p^{14} T^{2} \) |
| 59 | \( 1 + 4954816467613 T + p^{14} T^{2} \) |
| 61 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 67 | \( 1 + 11652648832405 T + p^{14} T^{2} \) |
| 71 | \( 1 - 14633687116307 T + p^{14} T^{2} \) |
| 73 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 79 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 83 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 89 | \( 1 + 68889823168417 T + p^{14} T^{2} \) |
| 97 | \( 1 - 68911099629215 T + p^{14} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.88548555504378258966854113605, −15.34889424288392848920080124965, −14.15364067490745431664172450693, −12.91916212939582220070254328702, −10.75988742334461778376489929962, −9.319132507906152340943483480990, −7.61617666453320400902485672995, −5.79536832218465334955958767270, −2.90121695774692269707827184293, −1.89040868250106846753560087550,
1.89040868250106846753560087550, 2.90121695774692269707827184293, 5.79536832218465334955958767270, 7.61617666453320400902485672995, 9.319132507906152340943483480990, 10.75988742334461778376489929962, 12.91916212939582220070254328702, 14.15364067490745431664172450693, 15.34889424288392848920080124965, 16.88548555504378258966854113605