| L(s) = 1 | − 2·3-s + 2·5-s + 6·7-s + 3·9-s + 2·11-s − 6·13-s − 4·15-s − 2·19-s − 12·21-s + 12·23-s + 3·25-s − 4·27-s − 10·29-s + 4·31-s − 4·33-s + 12·35-s + 2·37-s + 12·39-s + 6·41-s + 6·43-s + 6·45-s + 20·47-s + 16·49-s + 4·55-s + 4·57-s + 12·59-s + 4·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 2.26·7-s + 9-s + 0.603·11-s − 1.66·13-s − 1.03·15-s − 0.458·19-s − 2.61·21-s + 2.50·23-s + 3/5·25-s − 0.769·27-s − 1.85·29-s + 0.718·31-s − 0.696·33-s + 2.02·35-s + 0.328·37-s + 1.92·39-s + 0.937·41-s + 0.914·43-s + 0.894·45-s + 2.91·47-s + 16/7·49-s + 0.539·55-s + 0.529·57-s + 1.56·59-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.049784007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.049784007\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T - 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 32 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 56 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 72 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 16 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 92 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 114 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 104 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 168 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.358828168188679457268774325187, −8.328761330012303401628125071910, −7.45746371869388320442434644017, −7.37776275523501008851050789506, −7.10261496941497293184373385082, −6.91868543611799108073037920098, −6.12165620070890867583643427463, −5.87384596442732967011460762441, −5.38297948864944214045115863347, −5.35905127228373661277160862333, −4.74746767906813541435230605277, −4.70702086601384250192774259482, −4.14630576128860509960993824945, −3.88310350469084650567535850728, −2.90189722109224836749925520217, −2.49300375073771484602837124552, −2.05706948783891339377207549974, −1.68782608690726585425533945137, −0.931144616665847884703921901701, −0.77798017491722243677583252580,
0.77798017491722243677583252580, 0.931144616665847884703921901701, 1.68782608690726585425533945137, 2.05706948783891339377207549974, 2.49300375073771484602837124552, 2.90189722109224836749925520217, 3.88310350469084650567535850728, 4.14630576128860509960993824945, 4.70702086601384250192774259482, 4.74746767906813541435230605277, 5.35905127228373661277160862333, 5.38297948864944214045115863347, 5.87384596442732967011460762441, 6.12165620070890867583643427463, 6.91868543611799108073037920098, 7.10261496941497293184373385082, 7.37776275523501008851050789506, 7.45746371869388320442434644017, 8.328761330012303401628125071910, 8.358828168188679457268774325187
