| L(s) = 1 | + 6·3-s + 15·4-s + 27·9-s + 90·12-s + 161·16-s − 74·17-s + 324·23-s + 201·25-s + 108·27-s − 226·29-s + 405·36-s + 492·43-s + 966·48-s + 586·49-s − 444·51-s − 1.07e3·53-s − 1.27e3·61-s + 1.45e3·64-s − 1.11e3·68-s + 1.94e3·69-s + 1.20e3·75-s + 1.76e3·79-s + 405·81-s − 1.35e3·87-s + 4.86e3·92-s + 3.01e3·100-s + 858·101-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 15/8·4-s + 9-s + 2.16·12-s + 2.51·16-s − 1.05·17-s + 2.93·23-s + 1.60·25-s + 0.769·27-s − 1.44·29-s + 15/8·36-s + 1.74·43-s + 2.90·48-s + 1.70·49-s − 1.21·51-s − 2.78·53-s − 2.66·61-s + 2.84·64-s − 1.97·68-s + 3.39·69-s + 1.85·75-s + 2.51·79-s + 5/9·81-s − 1.67·87-s + 5.50·92-s + 3.01·100-s + 0.845·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.449222544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.449222544\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 15 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 201 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 586 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 p^{2} T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 37 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 12818 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 162 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 113 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 21166 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 101137 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 56617 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 246 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 5798 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 537 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 78982 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 635 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 560722 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 463574 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 130009 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 884 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 875250 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1372302 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 380542 T^{2} + p^{6} T^{4} \) |
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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
−10.79460748587518372048437793175, −10.60112324143922350474529930907, −9.728178961169051786925102325261, −9.259996534925932011471982368967, −8.949728984531394534499386743901, −8.626010499916792556473548319619, −7.73828231743449996037902114865, −7.59288940331421725512631197431, −6.98619977537798814530031605912, −6.95098959712692682577528235478, −6.19121548472192415221577139489, −5.84731868777668843860899627645, −4.77352743291811944145030170546, −4.74329879418714130111604751333, −3.46349867047418674221993607443, −3.31624187138806325052107449747, −2.61569970950581122404835494376, −2.29676760800215134062938286735, −1.50429643173402262795720732483, −0.905421738761332474586948828172,
0.905421738761332474586948828172, 1.50429643173402262795720732483, 2.29676760800215134062938286735, 2.61569970950581122404835494376, 3.31624187138806325052107449747, 3.46349867047418674221993607443, 4.74329879418714130111604751333, 4.77352743291811944145030170546, 5.84731868777668843860899627645, 6.19121548472192415221577139489, 6.95098959712692682577528235478, 6.98619977537798814530031605912, 7.59288940331421725512631197431, 7.73828231743449996037902114865, 8.626010499916792556473548319619, 8.949728984531394534499386743901, 9.259996534925932011471982368967, 9.728178961169051786925102325261, 10.60112324143922350474529930907, 10.79460748587518372048437793175
