| L(s) = 1 | − 2·2-s + 5·5-s − 2·7-s + 8·8-s − 10·10-s + 9·11-s + 16·13-s + 4·14-s − 16·16-s − 12·17-s − 134·19-s − 18·22-s − 30·23-s − 32·26-s + 45·29-s + 247·31-s + 24·34-s − 10·35-s − 248·37-s + 268·38-s + 40·40-s − 3·41-s − 80·43-s + 60·46-s − 36·47-s + 343·49-s − 972·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.447·5-s − 0.107·7-s + 0.353·8-s − 0.316·10-s + 0.246·11-s + 0.341·13-s + 0.0763·14-s − 1/4·16-s − 0.171·17-s − 1.61·19-s − 0.174·22-s − 0.271·23-s − 0.241·26-s + 0.288·29-s + 1.43·31-s + 0.121·34-s − 0.0482·35-s − 1.10·37-s + 1.14·38-s + 0.158·40-s − 0.0114·41-s − 0.283·43-s + 0.192·46-s − 0.111·47-s + 49-s − 2.51·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2323994278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2323994278\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T - 339 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 9 T - 1250 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 16 T - 1941 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 67 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 30 T - 11267 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 45 T - 22364 T^{2} - 45 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 247 T + 31218 T^{2} - 247 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 124 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 68912 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 80 T - 73107 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 36 T - 102527 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 486 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 249 T - 143378 T^{2} + 249 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T - 226881 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 322 T - 197079 T^{2} - 322 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 453 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 346 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 352 T - 369135 T^{2} - 352 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 204 T - 530171 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 729 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 716 T - 400017 T^{2} + 716 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.29878978531305185569541728261, −9.486561865093999448510032672392, −9.347047526250006138605400814329, −8.635699329289032773038358357261, −8.618570937095451239685859266106, −7.929486377145118618592716261723, −7.80519458748246404743869963617, −6.84927218353861007105476395591, −6.72109686513836596150030909690, −6.25397886831394779055501196128, −5.81985884371208729966713699079, −5.15225813483892924948970086189, −4.70147157211585081119672892256, −4.10504319313035109193282272965, −3.78961647115509477691930253472, −2.78621393457628079302206550815, −2.55069524831290646288321004840, −1.44860501267533388384833964606, −1.44180042706952354295626631336, −0.14655382678887646394638648107,
0.14655382678887646394638648107, 1.44180042706952354295626631336, 1.44860501267533388384833964606, 2.55069524831290646288321004840, 2.78621393457628079302206550815, 3.78961647115509477691930253472, 4.10504319313035109193282272965, 4.70147157211585081119672892256, 5.15225813483892924948970086189, 5.81985884371208729966713699079, 6.25397886831394779055501196128, 6.72109686513836596150030909690, 6.84927218353861007105476395591, 7.80519458748246404743869963617, 7.929486377145118618592716261723, 8.618570937095451239685859266106, 8.635699329289032773038358357261, 9.347047526250006138605400814329, 9.486561865093999448510032672392, 10.29878978531305185569541728261
