| L(s) = 1 | − 8.19e3·4-s − 1.17e6·7-s + 6.71e7·16-s + 1.22e9·25-s + 9.66e9·28-s − 1.84e10·31-s + 8.49e11·49-s − 5.49e11·64-s − 4.34e11·73-s − 8.60e12·79-s + 3.18e13·97-s − 1.00e13·100-s − 2.19e13·103-s − 7.91e13·112-s + 1.02e13·121-s + 1.51e14·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6.05e14·169-s + 173-s + ⋯ |
| L(s) = 1 | − 4-s − 3.78·7-s + 16-s + 1.00·25-s + 3.78·28-s − 3.74·31-s + 8.76·49-s − 64-s − 0.336·73-s − 3.98·79-s + 3.87·97-s − 1.00·100-s − 1.81·103-s − 3.78·112-s + 0.296·121-s + 3.74·124-s + 2·169-s − 3.79·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.7299628823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7299628823\) |
| \(L(\frac{15}{2})\) |
|
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 | $C_2$ | \( 1 + p^{13} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 1221423842 T^{2} + p^{26} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 589678 T + p^{13} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10248475517270 T^{2} + p^{26} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p^{13} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{13} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p^{13} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{13} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 16632711200643501650 T^{2} + p^{26} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 9244937530 T + p^{13} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{13} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{13} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p^{13} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{13} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + \)\(52\!\cdots\!54\)\( T^{2} + p^{26} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + \)\(18\!\cdots\!30\)\( T^{2} + p^{26} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p^{13} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p^{13} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{13} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 217442403346 T + p^{13} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4304871108730 T + p^{13} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + \)\(61\!\cdots\!26\)\( T^{2} + p^{26} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p^{13} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 15903369256322 T + p^{13} T^{2} )^{2} \) |
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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
−12.61972167770600869052107666116, −11.99195645297268611562372233633, −10.89575972058655108218030429062, −10.38195606006891764416280115355, −9.808266560315943533699231104753, −9.537701631836565897870145113347, −8.854251449591573246440051340966, −8.852932442963560641928712895966, −7.36214955415761576638754563717, −7.12001475738948724993854730469, −6.44761608084923141922539834698, −5.75352399530314521814516208513, −5.53333525753236612351535462059, −4.28920798996640552555072690413, −3.73561916643643271348861750259, −3.16987038669355979477005023311, −2.98368793279071160323432386385, −1.80560754763977968327583397991, −0.47493275479784707608643199854, −0.44502128647493359609363474635,
0.44502128647493359609363474635, 0.47493275479784707608643199854, 1.80560754763977968327583397991, 2.98368793279071160323432386385, 3.16987038669355979477005023311, 3.73561916643643271348861750259, 4.28920798996640552555072690413, 5.53333525753236612351535462059, 5.75352399530314521814516208513, 6.44761608084923141922539834698, 7.12001475738948724993854730469, 7.36214955415761576638754563717, 8.852932442963560641928712895966, 8.854251449591573246440051340966, 9.537701631836565897870145113347, 9.808266560315943533699231104753, 10.38195606006891764416280115355, 10.89575972058655108218030429062, 11.99195645297268611562372233633, 12.61972167770600869052107666116
