| L(s) = 1 | − 2-s − 11·4-s + 10·5-s − 20·7-s + 15·8-s − 10·10-s + 14·11-s − 26·13-s + 20·14-s + 61·16-s + 82·17-s − 140·19-s − 110·20-s − 14·22-s + 54·23-s + 75·25-s + 26·26-s + 220·28-s + 114·29-s + 42·31-s − 89·32-s − 82·34-s − 200·35-s − 288·37-s + 140·38-s + 150·40-s + 264·41-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 1.37·4-s + 0.894·5-s − 1.07·7-s + 0.662·8-s − 0.316·10-s + 0.383·11-s − 0.554·13-s + 0.381·14-s + 0.953·16-s + 1.16·17-s − 1.69·19-s − 1.22·20-s − 0.135·22-s + 0.489·23-s + 3/5·25-s + 0.196·26-s + 1.48·28-s + 0.729·29-s + 0.243·31-s − 0.491·32-s − 0.413·34-s − 0.965·35-s − 1.27·37-s + 0.597·38-s + 0.592·40-s + 1.00·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.674260248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.674260248\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 p^{2} T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 20 T + 514 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 14 T + 2286 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 82 T + 6594 T^{2} - 82 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 140 T + 17530 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 54 T + 14438 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 114 T + 51602 T^{2} - 114 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 42 T + 59870 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 288 T + 92054 T^{2} + 288 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 264 T + 155198 T^{2} - 264 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 364 T + 183910 T^{2} - 364 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 694 T + 296622 T^{2} - 694 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 236590 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 66 T + 245230 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 556 T + 332958 T^{2} + 556 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 214 T + 330078 T^{2} + 214 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1526 T + 1297566 T^{2} - 1526 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 906 T + 836210 T^{2} - 906 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1860 T + 1674110 T^{2} - 1860 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 342 T + 1012862 T^{2} - 342 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 924 T + 1584214 T^{2} - 924 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2282 T + 2932594 T^{2} - 2282 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.23708689630129869639262968990, −10.16007678497258416848040138995, −9.453182415801194335145543377030, −9.228904952930854805644638433010, −8.991134966544478618038903855184, −8.570142958088418501489947706375, −7.72738033595545392442749904125, −7.72311235888536687845922261721, −6.74870686963602566282704877627, −6.45358082402670296621347858510, −6.00666339535360165536077575194, −5.46724374949459889814117577783, −4.82143733063995315196854781137, −4.60555985735362427394838012437, −3.68710965335451114831134341459, −3.51695686296988823935695756126, −2.52404538325067973355246951900, −2.05845804611839947667075805901, −0.856807365455473710243565172990, −0.58001528079371500355756594979,
0.58001528079371500355756594979, 0.856807365455473710243565172990, 2.05845804611839947667075805901, 2.52404538325067973355246951900, 3.51695686296988823935695756126, 3.68710965335451114831134341459, 4.60555985735362427394838012437, 4.82143733063995315196854781137, 5.46724374949459889814117577783, 6.00666339535360165536077575194, 6.45358082402670296621347858510, 6.74870686963602566282704877627, 7.72311235888536687845922261721, 7.72738033595545392442749904125, 8.570142958088418501489947706375, 8.991134966544478618038903855184, 9.228904952930854805644638433010, 9.453182415801194335145543377030, 10.16007678497258416848040138995, 10.23708689630129869639262968990
