| L(s) = 1 | + 6·11-s + 8·13-s − 2·25-s + 4·37-s + 24·47-s + 2·49-s + 30·59-s + 16·61-s − 12·71-s − 22·73-s − 24·83-s + 26·97-s + 6·107-s + 8·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 48·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + ⋯ |
| L(s) = 1 | + 1.80·11-s + 2.21·13-s − 2/5·25-s + 0.657·37-s + 3.50·47-s + 2/7·49-s + 3.90·59-s + 2.04·61-s − 1.42·71-s − 2.57·73-s − 2.63·83-s + 2.63·97-s + 0.580·107-s + 0.766·109-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4.01·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.352979699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.352979699\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
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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
−10.03434297233564260252939438253, −9.178466955321563983536739355857, −8.935080117476293169730794564247, −8.840227991322439352832031397699, −8.386830115888109776412920370753, −7.929805824135421689408514952505, −7.14930385665870715393193873159, −7.09263305290586059048594295861, −6.60443059315058874371763263857, −5.93213462606578208720240308081, −5.84105446650629553414329101778, −5.52551304385686249930763676027, −4.51108644376645875118903825586, −4.17208758990055764977770608121, −3.75233956916435402674675679681, −3.58340207176620914205424490886, −2.65489835237128257274058284684, −2.09386641525624104325426519838, −1.14548917146197857190991517174, −1.00161296328331945010126619260,
1.00161296328331945010126619260, 1.14548917146197857190991517174, 2.09386641525624104325426519838, 2.65489835237128257274058284684, 3.58340207176620914205424490886, 3.75233956916435402674675679681, 4.17208758990055764977770608121, 4.51108644376645875118903825586, 5.52551304385686249930763676027, 5.84105446650629553414329101778, 5.93213462606578208720240308081, 6.60443059315058874371763263857, 7.09263305290586059048594295861, 7.14930385665870715393193873159, 7.929805824135421689408514952505, 8.386830115888109776412920370753, 8.840227991322439352832031397699, 8.935080117476293169730794564247, 9.178466955321563983536739355857, 10.03434297233564260252939438253
