| L(s) = 1 | − 2·7-s + 8·17-s − 16·23-s + 25-s + 14·31-s − 8·47-s − 11·49-s − 8·71-s + 30·73-s + 8·79-s − 24·89-s + 14·97-s + 8·103-s + 32·113-s − 16·119-s − 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 32·161-s + 163-s + 167-s + 10·169-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 1.94·17-s − 3.33·23-s + 1/5·25-s + 2.51·31-s − 1.16·47-s − 1.57·49-s − 0.949·71-s + 3.51·73-s + 0.900·79-s − 2.54·89-s + 1.42·97-s + 0.788·103-s + 3.01·113-s − 1.46·119-s − 0.272·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 2.52·161-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11943936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11943936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.886771118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.886771118\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 165 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 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
−8.409005127505659549667901572755, −8.403016936834538288469810724710, −8.121010662458316823886881077367, −7.75328421705054306003732659572, −7.41690682663416185895850672719, −6.79750842604430871873806447742, −6.40831491063383685968229180649, −6.27470084574632242375286561871, −5.73022778056249847735557680364, −5.61693557461496464021712863491, −4.87725056324764156445244084425, −4.57348364286840330302588909792, −4.14748623122166565950796400327, −3.57179595346358459896554092565, −3.24585859912924708263753260273, −3.00373317061764681996532197417, −2.11582966785992878882918906320, −1.93406576309082238562826725070, −1.06985127984863526346976272214, −0.46305241842986166583116479799,
0.46305241842986166583116479799, 1.06985127984863526346976272214, 1.93406576309082238562826725070, 2.11582966785992878882918906320, 3.00373317061764681996532197417, 3.24585859912924708263753260273, 3.57179595346358459896554092565, 4.14748623122166565950796400327, 4.57348364286840330302588909792, 4.87725056324764156445244084425, 5.61693557461496464021712863491, 5.73022778056249847735557680364, 6.27470084574632242375286561871, 6.40831491063383685968229180649, 6.79750842604430871873806447742, 7.41690682663416185895850672719, 7.75328421705054306003732659572, 8.121010662458316823886881077367, 8.403016936834538288469810724710, 8.409005127505659549667901572755
