| L(s) = 1 | − 2-s − 6·3-s + 4-s + 6·6-s + 14·7-s − 9·8-s + 27·9-s − 22·11-s − 6·12-s + 22·13-s − 14·14-s − 47·16-s − 116·17-s − 27·18-s + 102·19-s − 84·21-s + 22·22-s − 260·23-s + 54·24-s − 22·26-s − 108·27-s + 14·28-s − 196·29-s + 150·31-s + 103·32-s + 132·33-s + 116·34-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 1.15·3-s + 1/8·4-s + 0.408·6-s + 0.755·7-s − 0.397·8-s + 9-s − 0.603·11-s − 0.144·12-s + 0.469·13-s − 0.267·14-s − 0.734·16-s − 1.65·17-s − 0.353·18-s + 1.23·19-s − 0.872·21-s + 0.213·22-s − 2.35·23-s + 0.459·24-s − 0.165·26-s − 0.769·27-s + 0.0944·28-s − 1.25·29-s + 0.869·31-s + 0.568·32-s + 0.696·33-s + 0.585·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 p T + 2718 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 22 T + 4450 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 116 T + 9030 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 102 T + 13134 T^{2} - 102 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 260 T + 34734 T^{2} + 260 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 196 T + 20942 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 150 T + 36542 T^{2} - 150 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 96 T + 82550 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 176 T - 16914 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 p T + 171958 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 560 T + 248606 T^{2} + 560 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 326 T + 204138 T^{2} + 326 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 844 T + 474182 T^{2} + 844 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 204 T + 455006 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 104 T + 537670 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1670 T + 1384382 T^{2} - 1670 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 386 T + 152218 T^{2} - 386 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 888 T + 1007454 T^{2} + 888 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 928 T + 600710 T^{2} + 928 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 588 T + 1495334 T^{2} - 588 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 522 T + 291282 T^{2} + 522 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.22830229563661867924912411589, −9.866617179104076557473098780024, −9.292910589293167707581866849496, −9.151774616763100073637836451075, −8.130216543569479799823416786368, −8.117656687226005390987407264881, −7.65155649499516237211362880703, −6.84290374163458097002565157603, −6.58582852725826489533600907087, −6.09193953812086260274418914693, −5.56991007286448801804177232082, −5.15502211710477551068967189687, −4.40949664216371750983075889168, −4.28137017897688057816930304822, −3.39507048704496452027599025816, −2.46535011381250003962058324278, −1.92262791641916818510829055902, −1.22479267346348945102799469931, 0, 0,
1.22479267346348945102799469931, 1.92262791641916818510829055902, 2.46535011381250003962058324278, 3.39507048704496452027599025816, 4.28137017897688057816930304822, 4.40949664216371750983075889168, 5.15502211710477551068967189687, 5.56991007286448801804177232082, 6.09193953812086260274418914693, 6.58582852725826489533600907087, 6.84290374163458097002565157603, 7.65155649499516237211362880703, 8.117656687226005390987407264881, 8.130216543569479799823416786368, 9.151774616763100073637836451075, 9.292910589293167707581866849496, 9.866617179104076557473098780024, 10.22830229563661867924912411589
