| L(s) = 1 | − 2-s − 7-s + 8-s − 5·13-s + 14-s − 16-s + 6·17-s + 4·19-s + 9·23-s + 5·25-s + 5·26-s + 3·29-s − 5·31-s − 6·34-s + 4·37-s − 4·38-s + 6·41-s + 43-s − 9·46-s + 6·47-s − 5·50-s + 6·53-s − 56-s − 3·58-s + 3·59-s + 10·61-s + 5·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.377·7-s + 0.353·8-s − 1.38·13-s + 0.267·14-s − 1/4·16-s + 1.45·17-s + 0.917·19-s + 1.87·23-s + 25-s + 0.980·26-s + 0.557·29-s − 0.898·31-s − 1.02·34-s + 0.657·37-s − 0.648·38-s + 0.937·41-s + 0.152·43-s − 1.32·46-s + 0.875·47-s − 0.707·50-s + 0.824·53-s − 0.133·56-s − 0.393·58-s + 0.390·59-s + 1.28·61-s + 0.635·62-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
| \(L(1)\) |
\(\approx\) |
\(1.573551759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.573551759\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.743242842817118384640006959781, −9.674969587994882901293654091028, −9.171755942611424211770756360185, −9.098102595540342519353851101158, −8.302475695907972696196273570799, −7.987529908860288326745395590526, −7.58863795877584340054175491142, −7.22463415239619556220526618009, −6.73492461025122486180398814718, −6.56906186312817737476245355543, −5.56696314300874590683227925059, −5.30776344582632844516317934434, −5.07660052629493629384249205120, −4.45461716176814425859228656958, −3.64674318317982057235986999314, −3.40329535485862069066616526557, −2.54214352953604196856965020672, −2.37204653928302793420948559482, −1.00896177670568410504459182838, −0.834373697048632301878664770132,
0.834373697048632301878664770132, 1.00896177670568410504459182838, 2.37204653928302793420948559482, 2.54214352953604196856965020672, 3.40329535485862069066616526557, 3.64674318317982057235986999314, 4.45461716176814425859228656958, 5.07660052629493629384249205120, 5.30776344582632844516317934434, 5.56696314300874590683227925059, 6.56906186312817737476245355543, 6.73492461025122486180398814718, 7.22463415239619556220526618009, 7.58863795877584340054175491142, 7.987529908860288326745395590526, 8.302475695907972696196273570799, 9.098102595540342519353851101158, 9.171755942611424211770756360185, 9.674969587994882901293654091028, 9.743242842817118384640006959781
