| L(s) = 1 | + 2·3-s − 4·4-s + 3·9-s − 8·12-s + 12·16-s + 12·23-s + 4·27-s − 10·31-s − 12·36-s + 4·37-s − 24·47-s + 24·48-s + 49-s + 12·53-s − 32·64-s − 14·67-s + 24·69-s + 24·71-s + 5·81-s + 12·89-s − 48·92-s − 20·93-s + 14·97-s + 32·103-s − 16·108-s + 8·111-s + 12·113-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 2·4-s + 9-s − 2.30·12-s + 3·16-s + 2.50·23-s + 0.769·27-s − 1.79·31-s − 2·36-s + 0.657·37-s − 3.50·47-s + 3.46·48-s + 1/7·49-s + 1.64·53-s − 4·64-s − 1.71·67-s + 2.88·69-s + 2.84·71-s + 5/9·81-s + 1.27·89-s − 5.00·92-s − 2.07·93-s + 1.42·97-s + 3.15·103-s − 1.53·108-s + 0.759·111-s + 1.12·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.434497126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.434497126\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 71 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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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
−7.84495571967658136968987786635, −7.76357865325971731170085778214, −7.23101848487913550379006421996, −7.19825361007192020082447539829, −6.51624259884145501508399424656, −6.20434185196982205686477680657, −5.78732821326004924874305247251, −5.17320313734328174728782172749, −4.96567772897849934733157476693, −4.93456669450477172723548127750, −4.42499478430098730328242956817, −3.95105246935832604290183888064, −3.52771476383994467980489110503, −3.43902314102186214166062186668, −3.03279540016212287639724124646, −2.57433844123180510194380940696, −1.77518229809312458528007717675, −1.62093596251054465961962235363, −0.71915084784756630160055871820, −0.60642013327410562819168845036,
0.60642013327410562819168845036, 0.71915084784756630160055871820, 1.62093596251054465961962235363, 1.77518229809312458528007717675, 2.57433844123180510194380940696, 3.03279540016212287639724124646, 3.43902314102186214166062186668, 3.52771476383994467980489110503, 3.95105246935832604290183888064, 4.42499478430098730328242956817, 4.93456669450477172723548127750, 4.96567772897849934733157476693, 5.17320313734328174728782172749, 5.78732821326004924874305247251, 6.20434185196982205686477680657, 6.51624259884145501508399424656, 7.19825361007192020082447539829, 7.23101848487913550379006421996, 7.76357865325971731170085778214, 7.84495571967658136968987786635
