| L(s) = 1 | + 48·19-s − 25·25-s − 72·31-s + 26·37-s + 44·43-s − 168·61-s − 122·67-s + 240·73-s + 142·79-s + 120·103-s − 214·109-s + 121·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 146·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2.52·19-s − 25-s − 2.32·31-s + 0.702·37-s + 1.02·43-s − 2.75·61-s − 1.82·67-s + 3.28·73-s + 1.79·79-s + 1.16·103-s − 1.96·109-s + 121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.863·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.347013773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.347013773\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 37 T + p^{2} T^{2} )( 1 - 11 T + p^{2} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 13 T + p^{2} T^{2} )( 1 + 59 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 73 T + p^{2} T^{2} )( 1 + 47 T + p^{2} T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 47 T + p^{2} T^{2} )( 1 + 121 T + p^{2} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 13 T + p^{2} T^{2} )( 1 + 109 T + p^{2} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 143 T + p^{2} T^{2} )( 1 - 97 T + p^{2} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 131 T + p^{2} T^{2} )( 1 - 11 T + p^{2} T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 2 T + p^{2} T^{2} ) \) |
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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
−9.372874321539911495544234444276, −9.021855430725347594184867888449, −8.635962559913660474865126202426, −7.87383306872839341925955146537, −7.64302190266010094140639654727, −7.53930252631966524905037476341, −7.09027104094531396437496849805, −6.43912586020080692470438130122, −6.06006645853070827091989674891, −5.66417582983816668698682358728, −5.25364460173590076304739125569, −4.93285016373796985227948771536, −4.32597274926836457411164260124, −3.75899846222041912529921123342, −3.41640410232224839599560299586, −2.99601005040899425607602515593, −2.30185439677607706467032870872, −1.73887631733641882119473445627, −1.14668635492091861950034106394, −0.43687757343711019386293760251,
0.43687757343711019386293760251, 1.14668635492091861950034106394, 1.73887631733641882119473445627, 2.30185439677607706467032870872, 2.99601005040899425607602515593, 3.41640410232224839599560299586, 3.75899846222041912529921123342, 4.32597274926836457411164260124, 4.93285016373796985227948771536, 5.25364460173590076304739125569, 5.66417582983816668698682358728, 6.06006645853070827091989674891, 6.43912586020080692470438130122, 7.09027104094531396437496849805, 7.53930252631966524905037476341, 7.64302190266010094140639654727, 7.87383306872839341925955146537, 8.635962559913660474865126202426, 9.021855430725347594184867888449, 9.372874321539911495544234444276
