| L(s) = 1 | − 4·9-s + 12·11-s − 2·23-s − 8·25-s + 8·29-s + 12·37-s + 8·43-s + 20·53-s + 28·67-s − 8·71-s − 12·79-s + 7·81-s − 48·99-s + 8·107-s − 4·109-s − 20·113-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + ⋯ |
| L(s) = 1 | − 4/3·9-s + 3.61·11-s − 0.417·23-s − 8/5·25-s + 1.48·29-s + 1.97·37-s + 1.21·43-s + 2.74·53-s + 3.42·67-s − 0.949·71-s − 1.35·79-s + 7/9·81-s − 4.82·99-s + 0.773·107-s − 0.383·109-s − 1.88·113-s + 7.81·121-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 + 0.0783·163-s + 0.0773·167-s − 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81288256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81288256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.795422145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.795422145\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 176 T^{2} + p^{2} T^{4} \) |
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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
−7.998983480517910725966820225688, −7.59122848476229552698230932198, −7.00515392885740850976455634761, −6.90888454984780455819530417217, −6.45480685391256898523059999620, −6.24994161840606596813761689999, −5.91960971253250079521014850394, −5.60488435857741353500634434141, −5.35475993306710774381329846402, −4.53691542719270714687755338306, −4.22640356655379761505701271331, −4.16625481691014764256093697445, −3.59274952475296978065693738790, −3.53786811004747854664225621151, −2.61462144213150749887682427863, −2.60902881374815277796706267749, −1.92478566545654297314602336965, −1.47155296075684904983007223790, −0.819122145286882131617120176754, −0.68364278136733429383122331521,
0.68364278136733429383122331521, 0.819122145286882131617120176754, 1.47155296075684904983007223790, 1.92478566545654297314602336965, 2.60902881374815277796706267749, 2.61462144213150749887682427863, 3.53786811004747854664225621151, 3.59274952475296978065693738790, 4.16625481691014764256093697445, 4.22640356655379761505701271331, 4.53691542719270714687755338306, 5.35475993306710774381329846402, 5.60488435857741353500634434141, 5.91960971253250079521014850394, 6.24994161840606596813761689999, 6.45480685391256898523059999620, 6.90888454984780455819530417217, 7.00515392885740850976455634761, 7.59122848476229552698230932198, 7.998983480517910725966820225688
