| L(s) = 1 | + 2-s − 11·4-s + 10·5-s − 14·7-s − 15·8-s + 10·10-s − 4·11-s − 22·13-s − 14·14-s + 61·16-s − 58·17-s − 110·20-s − 4·22-s − 82·23-s + 75·25-s − 22·26-s + 154·28-s − 334·29-s − 210·31-s + 89·32-s − 58·34-s − 140·35-s + 6·37-s − 150·40-s − 176·41-s + 46·43-s + 44·44-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 1.37·4-s + 0.894·5-s − 0.755·7-s − 0.662·8-s + 0.316·10-s − 0.109·11-s − 0.469·13-s − 0.267·14-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 + 1.03·28-s − 2.13·29-s − 1.21·31-s + 0.491·32-s − 0.292·34-s − 0.676·35-s + 0.0266·37-s − 0.592·40-s − 0.670·41-s + 0.163·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 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 | $D_{4}$ | \( 1 - 1626 T + 1498938 T^{2} - 1626 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.83220766002518717995974441685, −10.57431655397137215471692263117, −9.697533450317179146007052011089, −9.662480775750861955170523573168, −9.120194146511698226614978741361, −9.018914884707597032364058327883, −8.098503659432543357784373585902, −7.78573627573184770330176879861, −6.97849430202312930360600406647, −6.52369794341279332217201960359, −5.76957120946380157339099050648, −5.64615198411599812420155516438, −4.68081441727888723927357170415, −4.65388526606627060183646159169, −3.56384232611935751687555889382, −3.37030543644790019091490755104, −2.24367500346555214169611067185, −1.60539406169862632853283362378, 0, 0,
1.60539406169862632853283362378, 2.24367500346555214169611067185, 3.37030543644790019091490755104, 3.56384232611935751687555889382, 4.65388526606627060183646159169, 4.68081441727888723927357170415, 5.64615198411599812420155516438, 5.76957120946380157339099050648, 6.52369794341279332217201960359, 6.97849430202312930360600406647, 7.78573627573184770330176879861, 8.098503659432543357784373585902, 9.018914884707597032364058327883, 9.120194146511698226614978741361, 9.662480775750861955170523573168, 9.697533450317179146007052011089, 10.57431655397137215471692263117, 10.83220766002518717995974441685
