| L(s) = 1 | − 20·5-s − 9·9-s + 92·11-s + 208·19-s + 275·25-s − 448·29-s + 144·31-s + 388·41-s + 180·45-s + 586·49-s − 1.84e3·55-s + 852·59-s + 1.39e3·61-s − 376·71-s + 2.33e3·79-s + 81·81-s − 2.41e3·89-s − 4.16e3·95-s − 828·99-s − 2.25e3·101-s − 3.20e3·109-s + 3.68e3·121-s − 3.00e3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1/3·9-s + 2.52·11-s + 2.51·19-s + 11/5·25-s − 2.86·29-s + 0.834·31-s + 1.47·41-s + 0.596·45-s + 1.70·49-s − 4.51·55-s + 1.88·59-s + 2.93·61-s − 0.628·71-s + 3.32·79-s + 1/9·81-s − 2.87·89-s − 4.49·95-s − 0.840·99-s − 2.22·101-s − 2.81·109-s + 2.76·121-s − 2.14·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.101857223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.101857223\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 + 4 p T + p^{3} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 586 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 46 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3238 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 5470 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 104 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 2562 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 224 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 72 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 100822 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 194 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 147350 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 22754 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 215958 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 426 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 698 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 493942 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 188 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 230434 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1168 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 973830 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1206 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 90110 T^{2} + p^{6} T^{4} \) |
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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
−11.82743919665936893472680894648, −11.56443050906327530517220662032, −11.21397825538864847938256684229, −10.77716432500795205941830858081, −9.727043493866489091855245491148, −9.446368007447710611995930170548, −9.104298777559835004724024263880, −8.475669382080289593855659949869, −7.922391735247120713232863085489, −7.44222762645074758291071728648, −6.96437411932091214153296068620, −6.65502269083394473260254694649, −5.54211626932459469915570398594, −5.38159035300773826298257328281, −4.17196759682810013289975334958, −3.84986752814880570112904772219, −3.62648118495801271238709323632, −2.62346782158293543362061476002, −1.25387175509316289668548395249, −0.70148799396917029260058376361,
0.70148799396917029260058376361, 1.25387175509316289668548395249, 2.62346782158293543362061476002, 3.62648118495801271238709323632, 3.84986752814880570112904772219, 4.17196759682810013289975334958, 5.38159035300773826298257328281, 5.54211626932459469915570398594, 6.65502269083394473260254694649, 6.96437411932091214153296068620, 7.44222762645074758291071728648, 7.922391735247120713232863085489, 8.475669382080289593855659949869, 9.104298777559835004724024263880, 9.446368007447710611995930170548, 9.727043493866489091855245491148, 10.77716432500795205941830858081, 11.21397825538864847938256684229, 11.56443050906327530517220662032, 11.82743919665936893472680894648
