| L(s) = 1 | − 2-s − 6·5-s − 4·7-s + 8-s + 6·10-s − 6·11-s + 4·13-s + 4·14-s − 16-s + 4·19-s + 6·22-s + 17·25-s − 4·26-s + 9·29-s + 31-s + 24·35-s − 8·37-s − 4·38-s − 6·40-s + 10·43-s − 6·47-s + 9·49-s − 17·50-s − 3·53-s + 36·55-s − 4·56-s − 9·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.68·5-s − 1.51·7-s + 0.353·8-s + 1.89·10-s − 1.80·11-s + 1.10·13-s + 1.06·14-s − 1/4·16-s + 0.917·19-s + 1.27·22-s + 17/5·25-s − 0.784·26-s + 1.67·29-s + 0.179·31-s + 4.05·35-s − 1.31·37-s − 0.648·38-s − 0.948·40-s + 1.52·43-s − 0.875·47-s + 9/7·49-s − 2.40·50-s − 0.412·53-s + 4.85·55-s − 0.534·56-s − 1.18·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3484194328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3484194328\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−10.11037854145630395391381806303, −9.653612908057771892577354172587, −9.015294700930763744891030780614, −8.744369563981517876634704711525, −8.205699823767514146455640557572, −8.075290885194319396111210317419, −7.68951186471539957170931728713, −7.28234333246632456654464533604, −6.90395084743673578452748678067, −6.43419274011781789983994184229, −5.89901429984915536176116827754, −5.14221082664712962715647889333, −4.91619895746372605971117492610, −4.13137034474540840853394773361, −3.81420342988359060376885150845, −3.28750892510202225909780605845, −3.07575075472159134007379977860, −2.31352085574566267005931063868, −0.867481232918765214860434098757, −0.41478198035576836475555357687,
0.41478198035576836475555357687, 0.867481232918765214860434098757, 2.31352085574566267005931063868, 3.07575075472159134007379977860, 3.28750892510202225909780605845, 3.81420342988359060376885150845, 4.13137034474540840853394773361, 4.91619895746372605971117492610, 5.14221082664712962715647889333, 5.89901429984915536176116827754, 6.43419274011781789983994184229, 6.90395084743673578452748678067, 7.28234333246632456654464533604, 7.68951186471539957170931728713, 8.075290885194319396111210317419, 8.205699823767514146455640557572, 8.744369563981517876634704711525, 9.015294700930763744891030780614, 9.653612908057771892577354172587, 10.11037854145630395391381806303
