L(s) = 1 | + 2·2-s − 9·5-s − 28·7-s − 8·8-s − 18·10-s − 57·11-s − 140·13-s − 56·14-s − 16·16-s + 51·17-s − 5·19-s − 114·22-s + 69·23-s + 125·25-s − 280·26-s − 228·29-s − 23·31-s + 102·34-s + 252·35-s + 253·37-s − 10·38-s + 72·40-s + 84·41-s − 248·43-s + 138·46-s + 201·47-s + 441·49-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.804·5-s − 1.51·7-s − 0.353·8-s − 0.569·10-s − 1.56·11-s − 2.98·13-s − 1.06·14-s − 1/4·16-s + 0.727·17-s − 0.0603·19-s − 1.10·22-s + 0.625·23-s + 25-s − 2.11·26-s − 1.45·29-s − 0.133·31-s + 0.514·34-s + 1.21·35-s + 1.12·37-s − 0.0426·38-s + 0.284·40-s + 0.319·41-s − 0.879·43-s + 0.442·46-s + 0.623·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15876 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15876 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.2877796024\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2877796024\) |
\(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 p T + p^{3} T^{2} \) |
good | 5 | $C_2^2$ | \( 1 + 9 T - 44 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 57 T + 1918 T^{2} + 57 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 70 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3 p T - 8 p^{2} T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T - 6834 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 p T - 14 p^{2} T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 114 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 23 T - 29262 T^{2} + 23 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 253 T + 13356 T^{2} - 253 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 42 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 124 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 201 T - 63422 T^{2} - 201 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 393 T + 5572 T^{2} + 393 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 219 T - 157418 T^{2} - 219 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 709 T + 275700 T^{2} - 709 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 419 T - 125202 T^{2} + 419 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 96 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 313 T - 291048 T^{2} - 313 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 461 T - 280518 T^{2} + 461 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 588 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 1017 T + 329320 T^{2} + 1017 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1834 T + p^{3} T^{2} )^{2} \) |
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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
−13.22193302360465146292285943151, −12.49961995031055694137648533451, −12.46708913244479249169247838330, −11.89633775663419552688758884824, −11.16927732145029875875820641765, −10.53335379446256411217460143678, −9.985276150510773421670946664228, −9.523755994453288507746170304785, −9.227248432905148834380697857327, −7.954615564962714627674081098571, −7.84654656569865350358548293331, −6.99941496234392294850665894798, −6.80027929810218148264468001856, −5.42471714336199981780341565789, −5.40557798676423664182083418600, −4.56403330061189237386154202555, −3.77568786941407795045817107590, −2.75199320760008547604671278558, −2.69233484650968199794323078482, −0.23283031929836453490315178253,
0.23283031929836453490315178253, 2.69233484650968199794323078482, 2.75199320760008547604671278558, 3.77568786941407795045817107590, 4.56403330061189237386154202555, 5.40557798676423664182083418600, 5.42471714336199981780341565789, 6.80027929810218148264468001856, 6.99941496234392294850665894798, 7.84654656569865350358548293331, 7.954615564962714627674081098571, 9.227248432905148834380697857327, 9.523755994453288507746170304785, 9.985276150510773421670946664228, 10.53335379446256411217460143678, 11.16927732145029875875820641765, 11.89633775663419552688758884824, 12.46708913244479249169247838330, 12.49961995031055694137648533451, 13.22193302360465146292285943151