| L(s) = 1 | − 2·2-s − 6·5-s − 7·7-s + 8·8-s + 12·10-s − 30·11-s + 106·13-s + 14·14-s − 16·16-s − 84·17-s + 97·19-s + 60·22-s + 84·23-s + 125·25-s − 212·26-s + 360·29-s − 179·31-s + 168·34-s + 42·35-s + 145·37-s − 194·38-s − 48·40-s − 252·41-s − 650·43-s − 168·46-s − 366·47-s − 294·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.536·5-s − 0.377·7-s + 0.353·8-s + 0.379·10-s − 0.822·11-s + 2.26·13-s + 0.267·14-s − 1/4·16-s − 1.19·17-s + 1.17·19-s + 0.581·22-s + 0.761·23-s + 25-s − 1.59·26-s + 2.30·29-s − 1.03·31-s + 0.847·34-s + 0.202·35-s + 0.644·37-s − 0.828·38-s − 0.189·40-s − 0.959·41-s − 2.30·43-s − 0.538·46-s − 1.13·47-s − 6/7·49-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
| \(L(2)\) |
\(\approx\) |
\(1.030363542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.030363542\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{3} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 6 T - 89 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 30 T - 431 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 53 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 84 T + 2143 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 97 T + 2550 T^{2} - 97 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 84 T - 5111 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 180 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 179 T + 2250 T^{2} + 179 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 145 T - 29628 T^{2} - 145 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 126 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 325 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 366 T + 30133 T^{2} + 366 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 768 T + 440947 T^{2} + 768 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 264 T - 135683 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 818 T + 442143 T^{2} + 818 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 523 T - 27234 T^{2} - 523 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 342 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 43 T - 387168 T^{2} - 43 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 1171 T + 878202 T^{2} - 1171 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 810 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 600 T - 344969 T^{2} + 600 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 386 T + p^{3} T^{2} )^{2} \) |
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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
−13.23631760972399076113305208127, −12.66417862033728567336785928234, −12.14110038804396856393985258513, −11.31224776686604811967310098881, −10.97029375619024527678803487266, −10.80006921978571499697118325398, −9.912975577999338355814206092532, −9.508046886470679068998888782265, −8.764141733591846507659728781155, −8.375620117802380598793634997412, −8.083101007813237997371852869290, −7.23929237673795837249758600062, −6.40915339964381588625444221209, −6.35924533062181775072023553217, −4.90463477642954608666329915241, −4.82475527453791270161547633151, −3.36612555773542702531939328324, −3.24119998891461775981616232013, −1.64331533452878247063824713395, −0.62679119020312122629829294592,
0.62679119020312122629829294592, 1.64331533452878247063824713395, 3.24119998891461775981616232013, 3.36612555773542702531939328324, 4.82475527453791270161547633151, 4.90463477642954608666329915241, 6.35924533062181775072023553217, 6.40915339964381588625444221209, 7.23929237673795837249758600062, 8.083101007813237997371852869290, 8.375620117802380598793634997412, 8.764141733591846507659728781155, 9.508046886470679068998888782265, 9.912975577999338355814206092532, 10.80006921978571499697118325398, 10.97029375619024527678803487266, 11.31224776686604811967310098881, 12.14110038804396856393985258513, 12.66417862033728567336785928234, 13.23631760972399076113305208127
