| L(s) = 1 | + 2-s − 11·4-s − 10·5-s − 15·8-s − 10·10-s − 4·11-s + 22·13-s + 61·16-s + 58·17-s + 110·20-s − 4·22-s − 82·23-s + 75·25-s + 22·26-s − 334·29-s + 210·31-s + 89·32-s + 58·34-s + 6·37-s + 150·40-s + 176·41-s + 46·43-s + 44·44-s − 82·46-s + 514·47-s + 75·50-s − 242·52-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 1.37·4-s − 0.894·5-s − 0.662·8-s − 0.316·10-s − 0.109·11-s + 0.469·13-s + 0.953·16-s + 0.827·17-s + 1.22·20-s − 0.0387·22-s − 0.743·23-s + 3/5·25-s + 0.165·26-s − 2.13·29-s + 1.21·31-s + 0.491·32-s + 0.292·34-s + 0.0266·37-s + 0.592·40-s + 0.670·41-s + 0.163·43-s + 0.150·44-s − 0.262·46-s + 1.59·47-s + 0.212·50-s − 0.645·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 p^{2} T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 2598 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 22 T + 2458 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 58 T + 7794 T^{2} - 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5490 T^{2} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 82 T + 25182 T^{2} + 82 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 334 T + 72842 T^{2} + 334 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 210 T + 69230 T^{2} - 210 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 97490 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 176 T + 134094 T^{2} - 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 46 T + 42430 T^{2} - 46 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 514 T + 269870 T^{2} - 514 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 808 T + 449478 T^{2} + 808 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 284 T + p^{3} T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 618 T + 515018 T^{2} - 618 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 694 T + 681118 T^{2} - 694 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 814 T + 769934 T^{2} + 814 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 82 T + 422290 T^{2} + 82 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 600 T + 618846 T^{2} - 600 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 268 T + 779030 T^{2} + 268 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 72 T + 448286 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1626 T + 1498938 T^{2} + 1626 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.313986404571865142652630448011, −8.282109135128132622762307548727, −7.75866819823711929078405531565, −7.52195641179090705481838590446, −7.02953864171686226132912444075, −6.52207133860112242905809928598, −5.89119935430835704372433845551, −5.79689757782884252790490275939, −5.11019922436989186607765222217, −5.01786645271672186444496826923, −4.29555929638007642409199067416, −4.03945007081128344518315577254, −3.69724288265100475036162569025, −3.41955156059658656429943430252, −2.66483754607630004393587062209, −2.20903851023028807854969532293, −1.23956327088675683827205019265, −0.944177348487544557330072115825, 0, 0,
0.944177348487544557330072115825, 1.23956327088675683827205019265, 2.20903851023028807854969532293, 2.66483754607630004393587062209, 3.41955156059658656429943430252, 3.69724288265100475036162569025, 4.03945007081128344518315577254, 4.29555929638007642409199067416, 5.01786645271672186444496826923, 5.11019922436989186607765222217, 5.79689757782884252790490275939, 5.89119935430835704372433845551, 6.52207133860112242905809928598, 7.02953864171686226132912444075, 7.52195641179090705481838590446, 7.75866819823711929078405531565, 8.282109135128132622762307548727, 8.313986404571865142652630448011
