| L(s) = 1 | + 2-s + 2·4-s + 7-s + 5·8-s − 3·11-s − 2·13-s + 14-s + 5·16-s + 7·17-s + 7·19-s − 3·22-s − 6·23-s + 5·25-s − 2·26-s + 2·28-s + 10·29-s + 10·32-s + 7·34-s − 8·37-s + 7·38-s + 4·43-s − 6·44-s − 6·46-s + 7·47-s − 6·49-s + 5·50-s − 4·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s + 0.377·7-s + 1.76·8-s − 0.904·11-s − 0.554·13-s + 0.267·14-s + 5/4·16-s + 1.69·17-s + 1.60·19-s − 0.639·22-s − 1.25·23-s + 25-s − 0.392·26-s + 0.377·28-s + 1.85·29-s + 1.76·32-s + 1.20·34-s − 1.31·37-s + 1.13·38-s + 0.609·43-s − 0.904·44-s − 0.884·46-s + 1.02·47-s − 6/7·49-s + 0.707·50-s − 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670761 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670761 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.586420384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.586420384\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−10.50062690503833975411734451573, −10.20705804844234247627631597015, −9.649093909263032705558068898876, −9.503742254310624357392569201554, −8.453963849240288350915207702426, −8.115862681939855909508179202790, −7.893982136723309033224020138270, −7.44277587955183467098704020207, −6.86628225970989970342813747361, −6.82243445376342445908359488603, −5.76145459283401314157428205993, −5.65876930971662911685572106022, −4.91294070431899775976157886807, −4.90515130100010930980228173606, −4.12992419497508574910374075878, −3.51361928136713848133171515249, −2.85778785863875008767717600319, −2.56480554037620395877932443742, −1.61074468775497046483449544064, −1.05941418290059272872830620394,
1.05941418290059272872830620394, 1.61074468775497046483449544064, 2.56480554037620395877932443742, 2.85778785863875008767717600319, 3.51361928136713848133171515249, 4.12992419497508574910374075878, 4.90515130100010930980228173606, 4.91294070431899775976157886807, 5.65876930971662911685572106022, 5.76145459283401314157428205993, 6.82243445376342445908359488603, 6.86628225970989970342813747361, 7.44277587955183467098704020207, 7.893982136723309033224020138270, 8.115862681939855909508179202790, 8.453963849240288350915207702426, 9.503742254310624357392569201554, 9.649093909263032705558068898876, 10.20705804844234247627631597015, 10.50062690503833975411734451573
