| L(s) = 1 | − 4-s + 6·5-s − 3·16-s − 6·20-s + 12·23-s + 17·25-s + 8·31-s − 22·37-s − 2·49-s + 18·53-s + 12·59-s + 7·64-s − 4·67-s + 12·71-s − 18·80-s − 18·89-s − 12·92-s − 14·97-s − 17·100-s + 28·103-s + 42·113-s + 72·115-s − 8·124-s + 18·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2.68·5-s − 3/4·16-s − 1.34·20-s + 2.50·23-s + 17/5·25-s + 1.43·31-s − 3.61·37-s − 2/7·49-s + 2.47·53-s + 1.56·59-s + 7/8·64-s − 0.488·67-s + 1.42·71-s − 2.01·80-s − 1.90·89-s − 1.25·92-s − 1.42·97-s − 1.69·100-s + 2.75·103-s + 3.95·113-s + 6.71·115-s − 0.718·124-s + 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.447865298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.447865298\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 79 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−10.02086063647710924178450116043, −9.842325386386337692590707857480, −9.216354959181625752243332828814, −8.870987129926948099475686342112, −8.596938216441731266076463871288, −8.450993844397007973929666407794, −7.26847517935432064212987032117, −7.00607633369541907559955104744, −6.81046356117150327321636010509, −6.19728146879401851309670428527, −5.76558471103732495983973628130, −5.37454660274404579404395867177, −4.87171789207207126244139650640, −4.86693552311694469802699538494, −3.83786711455443175585011229911, −3.27007684045341957162730112055, −2.55800580814670307665353109157, −2.21393848240163495721722246576, −1.59063782793091227817997843551, −0.873365374176186409069496759274,
0.873365374176186409069496759274, 1.59063782793091227817997843551, 2.21393848240163495721722246576, 2.55800580814670307665353109157, 3.27007684045341957162730112055, 3.83786711455443175585011229911, 4.86693552311694469802699538494, 4.87171789207207126244139650640, 5.37454660274404579404395867177, 5.76558471103732495983973628130, 6.19728146879401851309670428527, 6.81046356117150327321636010509, 7.00607633369541907559955104744, 7.26847517935432064212987032117, 8.450993844397007973929666407794, 8.596938216441731266076463871288, 8.870987129926948099475686342112, 9.216354959181625752243332828814, 9.842325386386337692590707857480, 10.02086063647710924178450116043
