L(s) = 1 | + 2·5-s + 4·7-s + 2·11-s − 8·13-s − 8·17-s + 3·25-s − 4·29-s + 8·35-s − 4·37-s − 12·41-s + 12·43-s − 2·49-s − 12·53-s + 4·55-s − 8·59-s + 4·61-s − 16·65-s − 8·67-s − 8·73-s + 8·77-s − 8·79-s − 12·83-s − 16·85-s + 4·89-s − 32·91-s − 4·97-s + 4·101-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 1.51·7-s + 0.603·11-s − 2.21·13-s − 1.94·17-s + 3/5·25-s − 0.742·29-s + 1.35·35-s − 0.657·37-s − 1.87·41-s + 1.82·43-s − 2/7·49-s − 1.64·53-s + 0.539·55-s − 1.04·59-s + 0.512·61-s − 1.98·65-s − 0.977·67-s − 0.936·73-s + 0.911·77-s − 0.900·79-s − 1.31·83-s − 1.73·85-s + 0.423·89-s − 3.35·91-s − 0.406·97-s + 0.398·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 154 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.51862740621545430320353853534, −7.39291328395012450288020043006, −6.99783575495615306952749849876, −6.69047448454400086278687629853, −6.25709363217033247251568889580, −6.05413838640723643893804521290, −5.29621773796912226187455187371, −5.27776512102457242011056553286, −4.79834693817511945426182034769, −4.69629695987051252579372220201, −4.07797875330893851796531796242, −4.05014062082837923559011409610, −2.99818717418737689917731284240, −2.93385558158188956855720097389, −2.20958152387170289383865167206, −2.11442569191754284853753991655, −1.46340549983227464053364995930, −1.41221950009749839239270445780, 0, 0,
1.41221950009749839239270445780, 1.46340549983227464053364995930, 2.11442569191754284853753991655, 2.20958152387170289383865167206, 2.93385558158188956855720097389, 2.99818717418737689917731284240, 4.05014062082837923559011409610, 4.07797875330893851796531796242, 4.69629695987051252579372220201, 4.79834693817511945426182034769, 5.27776512102457242011056553286, 5.29621773796912226187455187371, 6.05413838640723643893804521290, 6.25709363217033247251568889580, 6.69047448454400086278687629853, 6.99783575495615306952749849876, 7.39291328395012450288020043006, 7.51862740621545430320353853534