| L(s) = 1 | − 2·3-s − 4·4-s − 2·5-s − 3·9-s − 4·11-s + 8·12-s + 4·15-s + 12·16-s + 8·20-s + 16·23-s + 3·25-s + 14·27-s − 2·31-s + 8·33-s + 12·36-s + 2·37-s + 16·44-s + 6·45-s − 12·47-s − 24·48-s − 14·49-s + 10·53-s + 8·55-s + 22·59-s − 16·60-s − 32·64-s − 4·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2·4-s − 0.894·5-s − 9-s − 1.20·11-s + 2.30·12-s + 1.03·15-s + 3·16-s + 1.78·20-s + 3.33·23-s + 3/5·25-s + 2.69·27-s − 0.359·31-s + 1.39·33-s + 2·36-s + 0.328·37-s + 2.41·44-s + 0.894·45-s − 1.75·47-s − 3.46·48-s − 2·49-s + 1.37·53-s + 1.07·55-s + 2.86·59-s − 2.06·60-s − 4·64-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2907025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2907025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−7.31046477239498505884334187868, −6.93915603622663156044010585623, −6.55670658790506835470610536991, −5.82556015260130568883160226450, −5.44890211749766646861026772927, −5.17204724441945101354395936072, −4.92691899597547312483394760424, −4.66081150409801299621656311291, −3.95007292624245662048447678433, −3.31807421839422759406366977778, −3.14330660201336049402162930576, −2.53787621818954063248179958507, −1.09560787998188841749262372687, −0.62578924072786036405065008266, 0,
0.62578924072786036405065008266, 1.09560787998188841749262372687, 2.53787621818954063248179958507, 3.14330660201336049402162930576, 3.31807421839422759406366977778, 3.95007292624245662048447678433, 4.66081150409801299621656311291, 4.92691899597547312483394760424, 5.17204724441945101354395936072, 5.44890211749766646861026772927, 5.82556015260130568883160226450, 6.55670658790506835470610536991, 6.93915603622663156044010585623, 7.31046477239498505884334187868
