| L(s) = 1 | − 3·2-s − 4-s − 7·7-s + 9·8-s + 66·11-s + 11·13-s + 21·14-s − 9·16-s − 99·17-s − 77·19-s − 198·22-s − 33·23-s − 33·26-s + 7·28-s − 51·29-s + 43·31-s + 153·32-s + 297·34-s + 50·37-s + 231·38-s + 132·41-s − 88·43-s − 66·44-s + 99·46-s − 399·47-s − 575·49-s − 11·52-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 1/8·4-s − 0.377·7-s + 0.397·8-s + 1.80·11-s + 0.234·13-s + 0.400·14-s − 0.140·16-s − 1.41·17-s − 0.929·19-s − 1.91·22-s − 0.299·23-s − 0.248·26-s + 0.0472·28-s − 0.326·29-s + 0.249·31-s + 0.845·32-s + 1.49·34-s + 0.222·37-s + 0.986·38-s + 0.502·41-s − 0.312·43-s − 0.226·44-s + 0.317·46-s − 1.23·47-s − 1.67·49-s − 0.0293·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.199508342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.199508342\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + 3 T + 5 p T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + p T + 624 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 6 p T + 293 p T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 11 T + 2568 T^{2} - 11 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 99 T + 11608 T^{2} + 99 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 77 T + 9186 T^{2} + 77 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 33 T + 21628 T^{2} + 33 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 51 T + 49420 T^{2} + 51 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 43 T + 59970 T^{2} - 43 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 50 T + 77874 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 132 T + 62965 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 88 T + 160653 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 399 T + 179128 T^{2} + 399 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 54 T + 81970 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 798 T + 417367 T^{2} - 798 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 439 T + 411186 T^{2} - 439 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 988 T + 809625 T^{2} + 988 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1368 T + 1012606 T^{2} - 1368 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 455 T + 342636 T^{2} - 455 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 803 T + 359562 T^{2} + 803 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 813 T + 1214362 T^{2} + 813 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 396 T + 1406374 T^{2} + 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 736 T + 1874937 T^{2} + 736 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.789419606431602827700965303869, −8.723775388907963368157895885830, −8.388803382797914872596803430597, −8.154452444013911157413921237058, −7.34397465167248054743729491510, −7.11544292882494172155382072550, −6.56606916799766314166517330079, −6.33361504994928022538597555742, −6.13321551931219653972591991459, −5.47853716894662982062614011703, −4.66104450055483693054143952916, −4.63225852486862828658300664321, −4.00915965488359012619771582164, −3.71706231961849110151506649295, −3.13265975732687783245757982792, −2.49574185695734970553575731954, −1.82863684040038960179018369733, −1.56597719996452882934401440035, −0.62930670680123096850954197221, −0.44746934537890273910923762736,
0.44746934537890273910923762736, 0.62930670680123096850954197221, 1.56597719996452882934401440035, 1.82863684040038960179018369733, 2.49574185695734970553575731954, 3.13265975732687783245757982792, 3.71706231961849110151506649295, 4.00915965488359012619771582164, 4.63225852486862828658300664321, 4.66104450055483693054143952916, 5.47853716894662982062614011703, 6.13321551931219653972591991459, 6.33361504994928022538597555742, 6.56606916799766314166517330079, 7.11544292882494172155382072550, 7.34397465167248054743729491510, 8.154452444013911157413921237058, 8.388803382797914872596803430597, 8.723775388907963368157895885830, 8.789419606431602827700965303869
