| L(s) = 1 | − 3·2-s + 4-s − 10·5-s − 3·8-s + 30·10-s − 62·11-s + 6·13-s + 9·16-s + 40·17-s + 122·19-s − 10·20-s + 186·22-s − 16·23-s + 75·25-s − 18·26-s − 352·29-s − 66·31-s + 165·32-s − 120·34-s − 188·37-s − 366·38-s + 30·40-s + 16·41-s − 396·43-s − 62·44-s + 48·46-s − 188·47-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 1/8·4-s − 0.894·5-s − 0.132·8-s + 0.948·10-s − 1.69·11-s + 0.128·13-s + 9/64·16-s + 0.570·17-s + 1.47·19-s − 0.111·20-s + 1.80·22-s − 0.145·23-s + 3/5·25-s − 0.135·26-s − 2.25·29-s − 0.382·31-s + 0.911·32-s − 0.605·34-s − 0.835·37-s − 1.56·38-s + 0.118·40-s + 0.0609·41-s − 1.40·43-s − 0.212·44-s + 0.153·46-s − 0.583·47-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
| \(L(2)\) |
\(\approx\) |
\(0.5902238494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5902238494\) |
| \(L(\frac{5}{2})\) |
|
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 | $D_{4}$ | \( 1 + 3 T + p^{3} T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 62 T + 3582 T^{2} + 62 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 3378 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 40 T + 8750 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 122 T + 17398 T^{2} - 122 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 16 T + 34 p T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 352 T + 78278 T^{2} + 352 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 66 T + 45870 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 188 T + 44542 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 16 T + 18350 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 396 T + 2226 p T^{2} + 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 p T + 15582 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 982 T + 504354 T^{2} + 982 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 516 T + 418118 T^{2} - 516 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 880 T + 575238 T^{2} - 880 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 356 T + 100374 T^{2} + 356 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 310 T + 664038 T^{2} + 310 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 326 T + 789802 T^{2} + 326 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1832 T + 1824478 T^{2} - 1832 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 680 T + 744870 T^{2} + 680 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 796 T + 411158 T^{2} - 796 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 670 T + 1145410 T^{2} - 670 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.765023127653518868449475180324, −8.552390438446042517550388443818, −8.011307591654896696318463065031, −7.928121915751284233212196480082, −7.43039046950796777063564665707, −7.28162703920366389412844994007, −6.77500121526243522454268513281, −6.10789216281293128148667186853, −5.76951821820016139012997102054, −5.29085316505930329359099282207, −4.85458081491807854820364249910, −4.75093120025812212571984131523, −3.71303461658186779984528906303, −3.50589041345024107644173498813, −3.21303841623059107067077121610, −2.55841064261338600173888301568, −1.94340258328469866153911550303, −1.41669654627431476667269737316, −0.52963839797465127935562734935, −0.35838979906258974904439614630,
0.35838979906258974904439614630, 0.52963839797465127935562734935, 1.41669654627431476667269737316, 1.94340258328469866153911550303, 2.55841064261338600173888301568, 3.21303841623059107067077121610, 3.50589041345024107644173498813, 3.71303461658186779984528906303, 4.75093120025812212571984131523, 4.85458081491807854820364249910, 5.29085316505930329359099282207, 5.76951821820016139012997102054, 6.10789216281293128148667186853, 6.77500121526243522454268513281, 7.28162703920366389412844994007, 7.43039046950796777063564665707, 7.928121915751284233212196480082, 8.011307591654896696318463065031, 8.552390438446042517550388443818, 8.765023127653518868449475180324
