| L(s) = 1 | + 6·5-s + 19·25-s + 20·31-s + 10·49-s − 12·53-s + 40·79-s + 60·83-s − 60·107-s − 10·121-s + 42·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 120·155-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 2.68·5-s + 19/5·25-s + 3.59·31-s + 10/7·49-s − 1.64·53-s + 4.50·79-s + 6.58·83-s − 5.80·107-s − 0.909·121-s + 3.75·125-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 + 9.63·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.54971367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.54971367\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 6 T + 65 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 30 T + 383 T^{2} - 30 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.98164125927132612265884487472, −5.71784766038546335383510425193, −5.57123934158246321745529866395, −5.25244831611733812850402622516, −5.06438700631489712013046754584, −5.03249925164661464188541986496, −4.88713363627910998152398852501, −4.65965073601846134026916395668, −4.50946078762964627774566901667, −4.02925762100155927418124077467, −3.85441093565431295985370769664, −3.71710301467746601081274263795, −3.55525346482591274314836628573, −3.14679529054714693457418294591, −2.85798895599717833703514615525, −2.72499085609469701486428561142, −2.42759142656421376448140922481, −2.41374173686696653629809664309, −2.17123528714976401875446625028, −1.82100510959271837410108288437, −1.64569260894103388567602860367, −1.23698641649096964904934138472, −1.04334835271819314882486502151, −0.78629376689442543067410922033, −0.43944588299476600526373906489,
0.43944588299476600526373906489, 0.78629376689442543067410922033, 1.04334835271819314882486502151, 1.23698641649096964904934138472, 1.64569260894103388567602860367, 1.82100510959271837410108288437, 2.17123528714976401875446625028, 2.41374173686696653629809664309, 2.42759142656421376448140922481, 2.72499085609469701486428561142, 2.85798895599717833703514615525, 3.14679529054714693457418294591, 3.55525346482591274314836628573, 3.71710301467746601081274263795, 3.85441093565431295985370769664, 4.02925762100155927418124077467, 4.50946078762964627774566901667, 4.65965073601846134026916395668, 4.88713363627910998152398852501, 5.03249925164661464188541986496, 5.06438700631489712013046754584, 5.25244831611733812850402622516, 5.57123934158246321745529866395, 5.71784766038546335383510425193, 5.98164125927132612265884487472
