| L(s) = 1 | − 8·4-s + 16·5-s − 18·9-s − 24·11-s + 48·16-s + 56·19-s − 128·20-s + 26·25-s + 448·29-s + 16·31-s + 144·36-s − 288·41-s + 192·44-s − 288·45-s − 98·49-s − 384·55-s − 512·59-s − 912·61-s − 256·64-s − 1.96e3·71-s − 448·76-s + 448·79-s + 768·80-s + 243·81-s − 80·89-s + 896·95-s + 432·99-s + ⋯ |
| L(s) = 1 | − 4-s + 1.43·5-s − 2/3·9-s − 0.657·11-s + 3/4·16-s + 0.676·19-s − 1.43·20-s + 0.207·25-s + 2.86·29-s + 0.0926·31-s + 2/3·36-s − 1.09·41-s + 0.657·44-s − 0.954·45-s − 2/7·49-s − 0.941·55-s − 1.12·59-s − 1.91·61-s − 1/2·64-s − 3.28·71-s − 0.676·76-s + 0.638·79-s + 1.07·80-s + 1/3·81-s − 0.0952·89-s + 0.967·95-s + 0.438·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.01625356377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01625356377\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 16 T + 46 p T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 11 | $D_{4}$ | \( ( 1 + 12 T + 1354 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 428 T^{2} + 8323158 T^{4} - 428 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 3732 T^{2} + 20295398 T^{4} - 3732 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 28 T + 5514 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 10236 T^{2} - 41151898 T^{4} - 10236 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 224 T + 57206 T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 35322 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 155204 T^{2} + 10693572822 T^{4} - 155204 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 144 T + 118750 T^{2} + 144 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 149660 T^{2} + 18213997398 T^{4} - 149660 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 169524 T^{2} + 28738348838 T^{4} + 169524 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 263564 T^{2} + 41315828598 T^{4} - 263564 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 256 T + 292742 T^{2} + 256 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 456 T + 213542 T^{2} + 456 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 21700 T^{2} - 83331153162 T^{4} + 21700 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 984 T + 580810 T^{2} + 984 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 1476764 T^{2} + 846383359398 T^{4} - 1476764 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 224 T + 716046 T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 989036 T^{2} + 799488357462 T^{4} - 989036 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 40 T + 1391438 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 158236 T^{2} - 84748847802 T^{4} - 158236 p^{6} T^{6} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.531573344590516320027265488418, −8.341176543821371491966947023129, −8.310282702962232459473631959035, −7.64409001462231273293454499386, −7.55002889128208725509059018258, −7.39627614346733693180698473052, −6.93474756059210365853065720996, −6.31995658405282070312207019966, −6.26408209560378238230223438275, −6.12905566639893701644381612971, −5.94864723124221596545003554967, −5.29376823991481616178286800792, −5.14593167765251637282182366710, −4.89480304256768196601046958959, −4.82798584878994811954109720637, −4.13179153607396837746786202004, −4.05923869481015412275284006028, −3.31495900396130709301083832242, −3.12146957827705249853302875614, −2.69608329166519614258810054034, −2.52607601593386874341991894906, −1.82299278911081762043406374157, −1.32160384237932599394491304486, −1.06500562790745720517477535232, −0.02581808390018006303332081041,
0.02581808390018006303332081041, 1.06500562790745720517477535232, 1.32160384237932599394491304486, 1.82299278911081762043406374157, 2.52607601593386874341991894906, 2.69608329166519614258810054034, 3.12146957827705249853302875614, 3.31495900396130709301083832242, 4.05923869481015412275284006028, 4.13179153607396837746786202004, 4.82798584878994811954109720637, 4.89480304256768196601046958959, 5.14593167765251637282182366710, 5.29376823991481616178286800792, 5.94864723124221596545003554967, 6.12905566639893701644381612971, 6.26408209560378238230223438275, 6.31995658405282070312207019966, 6.93474756059210365853065720996, 7.39627614346733693180698473052, 7.55002889128208725509059018258, 7.64409001462231273293454499386, 8.310282702962232459473631959035, 8.341176543821371491966947023129, 8.531573344590516320027265488418
