| L(s) = 1 | + 2.18e3·3-s − 1.63e4·4-s + 1.46e6·7-s − 3.58e7·12-s − 2.42e8·13-s + 1.58e9·19-s + 3.19e9·21-s − 6.10e9·25-s − 1.04e10·27-s − 2.39e10·28-s − 5.47e10·31-s + 7.71e10·37-s − 5.31e11·39-s − 7.51e11·43-s + 1.45e12·49-s + 3.98e12·52-s + 3.45e12·57-s − 6.21e12·61-s + 4.39e12·64-s − 7.62e12·67-s + 2.05e13·73-s − 1.33e13·75-s − 2.59e13·76-s + 1.79e13·79-s − 2.28e13·81-s − 5.23e13·84-s − 3.54e14·91-s + ⋯ |
| L(s) = 1 | + 3-s − 4-s + 1.77·7-s − 12-s − 3.87·13-s + 1.76·19-s + 1.77·21-s − 25-s − 27-s − 1.77·28-s − 1.99·31-s + 0.813·37-s − 3.87·39-s − 2.76·43-s + 2.14·49-s + 3.87·52-s + 1.76·57-s − 1.97·61-s + 64-s − 1.25·67-s + 1.86·73-s − 75-s − 1.76·76-s + 0.937·79-s − 81-s − 1.77·84-s − 6.86·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+7)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(0.9324544142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9324544142\) |
| \(L(8)\) |
|
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 | $C_2$ | \( 1 - p^{7} T + p^{14} T^{2} \) |
| 7 | $C_2$ | \( 1 - 1461083 T + p^{14} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 121460953 T + p^{14} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 1512529226 T + p^{14} T^{2} )( 1 - 69090803 T + p^{14} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 23014293166 T + p^{14} T^{2} )( 1 + 31777822381 T + p^{14} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 188820174863 T + p^{14} T^{2} )( 1 + 111626070166 T + p^{14} T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 375951067189 T + p^{14} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2292094939633 T + p^{14} T^{2} )( 1 + 3922504906441 T + p^{14} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4349633776379 T + p^{14} T^{2} )( 1 + 11973142765462 T + p^{14} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17278443497906 T + p^{14} T^{2} )( 1 - 3286657037807 T + p^{14} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 38383122173618 T + p^{14} T^{2} )( 1 + 20384167559989 T + p^{14} T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 11651694897022 T + p^{14} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.83215135356337646863795034251, −14.55052987733822652218632969001, −13.96000829728448000187573160481, −13.55834315216361054143420088447, −12.49201118215143776327394975258, −11.86848389838022376642984449236, −11.39827862870058565008865776916, −10.12740420026549045311705981592, −9.486815041275943345771137771842, −9.216307836780991248937985454407, −8.145085907925370164012776881926, −7.58747180509224512014421586391, −7.32786783215246874125049580779, −5.22904328605593808585319092764, −5.12167623335907462726605213493, −4.38740147548638926545377930031, −3.22678512779489445760984960249, −2.27371993257418845985672106253, −1.76092814123771531956549913328, −0.28670503647325085913097830634,
0.28670503647325085913097830634, 1.76092814123771531956549913328, 2.27371993257418845985672106253, 3.22678512779489445760984960249, 4.38740147548638926545377930031, 5.12167623335907462726605213493, 5.22904328605593808585319092764, 7.32786783215246874125049580779, 7.58747180509224512014421586391, 8.145085907925370164012776881926, 9.216307836780991248937985454407, 9.486815041275943345771137771842, 10.12740420026549045311705981592, 11.39827862870058565008865776916, 11.86848389838022376642984449236, 12.49201118215143776327394975258, 13.55834315216361054143420088447, 13.96000829728448000187573160481, 14.55052987733822652218632969001, 14.83215135356337646863795034251
