| L(s) = 1 | − 8·2-s + 34·4-s − 10·5-s − 96·8-s + 80·10-s + 14·11-s − 50·13-s + 196·16-s − 50·17-s − 36·19-s − 340·20-s − 112·22-s − 244·23-s + 75·25-s + 400·26-s + 26·29-s + 120·31-s − 352·32-s + 400·34-s + 564·37-s + 288·38-s + 960·40-s − 328·41-s − 260·43-s + 476·44-s + 1.95e3·46-s − 350·47-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 17/4·4-s − 0.894·5-s − 4.24·8-s + 2.52·10-s + 0.383·11-s − 1.06·13-s + 3.06·16-s − 0.713·17-s − 0.434·19-s − 3.80·20-s − 1.08·22-s − 2.21·23-s + 3/5·25-s + 3.01·26-s + 0.166·29-s + 0.695·31-s − 1.94·32-s + 2.01·34-s + 2.50·37-s + 1.22·38-s + 3.79·40-s − 1.24·41-s − 0.922·43-s + 1.63·44-s + 6.25·46-s − 1.08·47-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
| \(L(2)\) |
\(\approx\) |
\(0.1105415692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1105415692\) |
| \(L(\frac{5}{2})\) |
|
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 | $D_{4}$ | \( 1 + p^{3} T + 15 p T^{2} + p^{6} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 14 T + 663 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 50 T + 4987 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 50 T + 387 p T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 36 T + 10170 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 244 T + 29970 T^{2} + 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 26 T + 47795 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 120 T - 1618 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 564 T + 173630 T^{2} - 564 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 p T + 133986 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 260 T + 166666 T^{2} + 260 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 350 T + 203423 T^{2} + 350 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 56 T + 265770 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 616 T + p^{3} T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 336 T + 458858 T^{2} + 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 152 T + 599110 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 952 T + p^{3} T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 676 T + 655606 T^{2} + 676 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1014 T + 1120119 T^{2} - 1014 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 376 T + 458918 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 216 T + 1417730 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2742 T + 3608187 T^{2} + 2742 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.629477165124395765601450922261, −8.627095011600084465911495528876, −8.130389573449330633981585808386, −7.974382148278485253454901521137, −7.52334533966483751776965494171, −7.41506547769814586865968300835, −6.58357001980543113333025846270, −6.55818517723581852185939076514, −6.17393941239690889513943197217, −5.47405433000006111817361783440, −4.64105641687004024537122502259, −4.63555823797055523460087386838, −3.90853325173993302794829625465, −3.47761927849372547556817223125, −2.61926956576571861635333889769, −2.44155890006596279466748804477, −1.63899887474237095254948433205, −1.44012543751712632197422985552, −0.55157203883921794976160706921, −0.19871677461344758022014284013,
0.19871677461344758022014284013, 0.55157203883921794976160706921, 1.44012543751712632197422985552, 1.63899887474237095254948433205, 2.44155890006596279466748804477, 2.61926956576571861635333889769, 3.47761927849372547556817223125, 3.90853325173993302794829625465, 4.63555823797055523460087386838, 4.64105641687004024537122502259, 5.47405433000006111817361783440, 6.17393941239690889513943197217, 6.55818517723581852185939076514, 6.58357001980543113333025846270, 7.41506547769814586865968300835, 7.52334533966483751776965494171, 7.974382148278485253454901521137, 8.130389573449330633981585808386, 8.627095011600084465911495528876, 8.629477165124395765601450922261
