| L(s) = 1 | − 2·7-s + 16·11-s − 82·13-s + 8·17-s − 210·19-s + 40·23-s + 240·29-s + 218·31-s − 364·37-s − 84·41-s − 274·43-s + 524·47-s − 247·49-s + 444·53-s + 1.08e3·59-s + 774·61-s − 210·67-s − 1.10e3·71-s − 492·73-s − 32·77-s − 1.32e3·79-s − 28·83-s − 1.42e3·89-s + 164·91-s − 2.37e3·97-s − 828·101-s − 1.93e3·103-s + ⋯ |
| L(s) = 1 | − 0.107·7-s + 0.438·11-s − 1.74·13-s + 0.114·17-s − 2.53·19-s + 0.362·23-s + 1.53·29-s + 1.26·31-s − 1.61·37-s − 0.319·41-s − 0.971·43-s + 1.62·47-s − 0.720·49-s + 1.15·53-s + 2.39·59-s + 1.62·61-s − 0.382·67-s − 1.85·71-s − 0.788·73-s − 0.0473·77-s − 1.89·79-s − 0.0370·83-s − 1.69·89-s + 0.188·91-s − 2.48·97-s − 0.815·101-s − 1.85·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4641837706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4641837706\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T + 251 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 16 T - 1198 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 82 T + 4331 T^{2} + 82 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T - 1058 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 210 T + 24307 T^{2} + 210 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 40 T + 24298 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 240 T + 62742 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 218 T + 36147 T^{2} - 218 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 364 T + 90830 T^{2} + 364 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 84 T + 135682 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 274 T + 3881 p T^{2} + 274 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 524 T + 213506 T^{2} - 524 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 444 T + 343114 T^{2} - 444 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1084 T + 676618 T^{2} - 1084 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 774 T + 596755 T^{2} - 774 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 210 T + 15667 T^{2} + 210 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1108 T + 656062 T^{2} + 1108 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 492 T + 822854 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1328 T + 1085150 T^{2} + 1328 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 28 T + 697306 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 16 p T + 1909906 T^{2} + 16 p^{4} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2370 T + 3117955 T^{2} + 2370 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.876416248451656339527629278693, −8.658648265088365022292382391090, −8.371556962720652617440891224682, −8.224136190557681688967731877255, −7.31853742830552253825849186505, −7.00155467051111203645796075832, −6.87153446376181274647469534185, −6.50650561418108627650056580061, −5.78530363004666736488681320534, −5.58384684856806899330253891162, −4.91520662442032142394850981153, −4.61764105235841312124844138776, −4.11780956899534261542317223276, −3.91772541451029654451580841098, −2.95271609250383857806876851806, −2.68952339007087121421590914132, −2.25091003714032173569063307091, −1.64183514423778633442674992939, −0.959590289230004109798269390635, −0.15946906841977142730612401996,
0.15946906841977142730612401996, 0.959590289230004109798269390635, 1.64183514423778633442674992939, 2.25091003714032173569063307091, 2.68952339007087121421590914132, 2.95271609250383857806876851806, 3.91772541451029654451580841098, 4.11780956899534261542317223276, 4.61764105235841312124844138776, 4.91520662442032142394850981153, 5.58384684856806899330253891162, 5.78530363004666736488681320534, 6.50650561418108627650056580061, 6.87153446376181274647469534185, 7.00155467051111203645796075832, 7.31853742830552253825849186505, 8.224136190557681688967731877255, 8.371556962720652617440891224682, 8.658648265088365022292382391090, 8.876416248451656339527629278693
