| L(s) = 1 | + 4·5-s + 6·7-s − 9-s − 2·11-s − 8·13-s + 6·17-s + 6·19-s − 2·23-s + 11·25-s − 6·29-s + 24·35-s − 16·37-s + 12·43-s − 4·45-s + 10·47-s + 18·49-s − 8·55-s + 14·59-s + 18·61-s − 6·63-s − 32·65-s + 4·67-s − 16·71-s − 10·73-s − 12·77-s + 81-s + 24·85-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 2.26·7-s − 1/3·9-s − 0.603·11-s − 2.21·13-s + 1.45·17-s + 1.37·19-s − 0.417·23-s + 11/5·25-s − 1.11·29-s + 4.05·35-s − 2.63·37-s + 1.82·43-s − 0.596·45-s + 1.45·47-s + 18/7·49-s − 1.07·55-s + 1.82·59-s + 2.30·61-s − 0.755·63-s − 3.96·65-s + 0.488·67-s − 1.89·71-s − 1.17·73-s − 1.36·77-s + 1/9·81-s + 2.60·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.621110092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.621110092\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−9.305544270788153867716460691144, −9.160657301248868099541947869520, −8.661434268012685585114832918393, −8.100685204909060588732973332020, −7.88297934297539585997064463765, −7.42872330630797786371914978531, −7.05574702625220604747094122323, −6.95937974214477581191853835304, −5.73536268868617346526658515072, −5.68600263342161129838902229823, −5.42246533468340576881487914935, −5.09952911585532438070048549509, −4.81867831764982969756041794381, −4.17050958103818754166343121366, −3.49478586234522662346977366213, −2.77307775471206553594532045148, −2.36758096512261307990923491622, −1.99661631922278832715255511633, −1.50756774601316574163663389881, −0.798783899016046029604492539806,
0.798783899016046029604492539806, 1.50756774601316574163663389881, 1.99661631922278832715255511633, 2.36758096512261307990923491622, 2.77307775471206553594532045148, 3.49478586234522662346977366213, 4.17050958103818754166343121366, 4.81867831764982969756041794381, 5.09952911585532438070048549509, 5.42246533468340576881487914935, 5.68600263342161129838902229823, 5.73536268868617346526658515072, 6.95937974214477581191853835304, 7.05574702625220604747094122323, 7.42872330630797786371914978531, 7.88297934297539585997064463765, 8.100685204909060588732973332020, 8.661434268012685585114832918393, 9.160657301248868099541947869520, 9.305544270788153867716460691144
