| L(s) = 1 | + 2·2-s + 3·4-s + 2·5-s + 4·7-s + 4·8-s + 4·10-s − 5·11-s − 6·13-s + 8·14-s + 5·16-s + 4·17-s + 4·19-s + 6·20-s − 10·22-s + 4·23-s + 5·25-s − 12·26-s + 12·28-s − 7·29-s + 6·31-s + 6·32-s + 8·34-s + 8·35-s − 8·37-s + 8·38-s + 8·40-s + 6·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.894·5-s + 1.51·7-s + 1.41·8-s + 1.26·10-s − 1.50·11-s − 1.66·13-s + 2.13·14-s + 5/4·16-s + 0.970·17-s + 0.917·19-s + 1.34·20-s − 2.13·22-s + 0.834·23-s + 25-s − 2.35·26-s + 2.26·28-s − 1.29·29-s + 1.07·31-s + 1.06·32-s + 1.37·34-s + 1.35·35-s − 1.31·37-s + 1.29·38-s + 1.26·40-s + 0.937·41-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
| \(L(1)\) |
\(\approx\) |
\(7.684767384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.684767384\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 7 T - 34 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.15965768758270726263802344904, −9.751629429527716385927317561787, −9.351765909098496548995232739890, −8.708561073442556179650331164240, −8.085853094283782705251715945283, −7.997528723035342657381666802011, −7.28294390040074188244438674279, −7.13447085635539441241188271660, −6.84759891320165157575489765626, −5.77583534029149056654467104280, −5.52911816155775780218357199961, −5.40545775814953363390883855875, −4.87778445035891662310190805752, −4.71931544280298242307784565299, −3.96789110106649266905388447973, −3.30981575671157054021274517671, −2.55823371600499328904540783514, −2.52422048446923244539279723219, −1.79510431774678117233425808359, −1.00973336661073530968017660656,
1.00973336661073530968017660656, 1.79510431774678117233425808359, 2.52422048446923244539279723219, 2.55823371600499328904540783514, 3.30981575671157054021274517671, 3.96789110106649266905388447973, 4.71931544280298242307784565299, 4.87778445035891662310190805752, 5.40545775814953363390883855875, 5.52911816155775780218357199961, 5.77583534029149056654467104280, 6.84759891320165157575489765626, 7.13447085635539441241188271660, 7.28294390040074188244438674279, 7.997528723035342657381666802011, 8.085853094283782705251715945283, 8.708561073442556179650331164240, 9.351765909098496548995232739890, 9.751629429527716385927317561787, 10.15965768758270726263802344904
