| L(s) = 1 | − 12·2-s + 64·4-s − 315·5-s − 686·7-s − 576·8-s + 3.78e3·10-s + 1.47e3·11-s + 8.23e3·14-s + 6.91e3·16-s + 5.22e3·17-s + 1.19e4·19-s − 2.01e4·20-s − 1.77e4·22-s − 5.91e3·23-s + 5.05e4·25-s − 4.39e4·28-s − 7.95e3·29-s − 2.21e4·31-s − 3.68e4·32-s − 6.27e4·34-s + 2.16e5·35-s + 6.15e4·37-s − 1.42e5·38-s + 1.81e5·40-s − 3.48e4·43-s + 9.46e4·44-s + 7.09e4·46-s + ⋯ |
| L(s) = 1 | − 3/2·2-s + 4-s − 2.51·5-s − 2·7-s − 9/8·8-s + 3.77·10-s + 1.11·11-s + 3·14-s + 1.68·16-s + 1.06·17-s + 1.73·19-s − 2.51·20-s − 1.66·22-s − 0.485·23-s + 3.23·25-s − 2·28-s − 0.326·29-s − 0.745·31-s − 9/8·32-s − 1.59·34-s + 5.03·35-s + 1.21·37-s − 2.60·38-s + 2.83·40-s − 0.438·43-s + 1.11·44-s + 0.728·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.3980296664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3980296664\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 p^{2} T + 5 p^{4} T^{2} + 3 p^{8} T^{3} + p^{12} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 63 p T + 1948 p^{2} T^{2} + 63 p^{7} T^{3} + p^{12} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 1479 T + 415880 T^{2} - 1479 p^{6} T^{3} + p^{12} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 9418418 T^{2} + p^{12} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 5229 T + 33251716 T^{2} - 5229 p^{6} T^{3} + p^{12} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 11907 T + 94304764 T^{2} - 11907 p^{6} T^{3} + p^{12} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 5913 T - 113072320 T^{2} + 5913 p^{6} T^{3} + p^{12} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3978 T + p^{6} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 22197 T + 1051739284 T^{2} + 22197 p^{6} T^{3} + p^{12} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 61577 T + 1226000520 T^{2} - 61577 p^{6} T^{3} + p^{12} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 2726428318 T^{2} + p^{12} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 17414 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 53109 T + 11719403956 T^{2} - 53109 p^{6} T^{3} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 60513 T - 18502537960 T^{2} + 60513 p^{6} T^{3} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 373653 T + 88719388444 T^{2} - 373653 p^{6} T^{3} + p^{12} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 281883 T + 78006382924 T^{2} - 281883 p^{6} T^{3} + p^{12} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 268777 T - 18217306440 T^{2} - 268777 p^{6} T^{3} + p^{12} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 101922 T + p^{6} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 550179 T + 252233203636 T^{2} - 550179 p^{6} T^{3} + p^{12} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 362231 T - 111876158160 T^{2} + 362231 p^{6} T^{3} + p^{12} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 606885669938 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2311533 T + 2278042894324 T^{2} - 2311533 p^{6} T^{3} + p^{12} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 626523106942 T^{2} + p^{12} T^{4} \) |
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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
−14.34963545245706284224031292325, −13.17006697111538048089198297608, −12.64373345049078606426795020306, −11.89708250708182099840510529197, −11.86613171560798112239510907754, −11.37325435662897219311907348316, −10.27160597846059796390011380630, −9.875469964125696764396769346250, −9.171543571106060066802124567160, −8.998473721598464354593354465028, −7.922100225694134851695399254684, −7.76738744420237876769704572930, −7.02434071419374814316981860964, −6.42171832856460709226162068612, −5.47905694294126829380028430448, −3.75921605741168136011107076043, −3.73267682614220158575535845778, −2.96684403353166086335265013419, −0.73485743619374839299270418305, −0.56749687417866988727752900766,
0.56749687417866988727752900766, 0.73485743619374839299270418305, 2.96684403353166086335265013419, 3.73267682614220158575535845778, 3.75921605741168136011107076043, 5.47905694294126829380028430448, 6.42171832856460709226162068612, 7.02434071419374814316981860964, 7.76738744420237876769704572930, 7.922100225694134851695399254684, 8.998473721598464354593354465028, 9.171543571106060066802124567160, 9.875469964125696764396769346250, 10.27160597846059796390011380630, 11.37325435662897219311907348316, 11.86613171560798112239510907754, 11.89708250708182099840510529197, 12.64373345049078606426795020306, 13.17006697111538048089198297608, 14.34963545245706284224031292325
