| L(s) = 1 | − 2·3-s − 4-s + 4·5-s + 2·7-s + 2·9-s + 2·12-s − 8·15-s + 16-s + 8·17-s + 10·19-s − 4·20-s − 4·21-s + 11·25-s − 6·27-s − 2·28-s + 6·29-s + 14·31-s + 8·35-s − 2·36-s − 12·37-s + 8·45-s − 14·47-s − 2·48-s + 2·49-s − 16·51-s + 2·53-s − 20·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 1.78·5-s + 0.755·7-s + 2/3·9-s + 0.577·12-s − 2.06·15-s + 1/4·16-s + 1.94·17-s + 2.29·19-s − 0.894·20-s − 0.872·21-s + 11/5·25-s − 1.15·27-s − 0.377·28-s + 1.11·29-s + 2.51·31-s + 1.35·35-s − 1/3·36-s − 1.97·37-s + 1.19·45-s − 2.04·47-s − 0.288·48-s + 2/7·49-s − 2.24·51-s + 0.274·53-s − 2.64·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.719538640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.719538640\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 37 | $C_2$ | \( 1 + 12 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−11.85802598157211148736903372086, −11.28381770311072609603543422432, −10.34648853603495862491516881824, −10.31116028768085837893689030303, −9.819277877016164792075999364668, −9.743408199867838337470935935711, −8.823461997305896704784924605521, −8.585083425982287523215195122926, −7.67713971542231896259485145227, −7.51897958150786303110733817438, −6.68526356117287462089483685564, −6.13613439725025295932208442071, −5.77520787672965429489541320677, −5.15287970719384896425515269860, −5.14830422622777510526286974851, −4.47664800244323080901804971475, −3.28734478069087000181489345612, −2.86806587891364778727409853097, −1.45751200082893960204319637478, −1.22265141455069238326682312772,
1.22265141455069238326682312772, 1.45751200082893960204319637478, 2.86806587891364778727409853097, 3.28734478069087000181489345612, 4.47664800244323080901804971475, 5.14830422622777510526286974851, 5.15287970719384896425515269860, 5.77520787672965429489541320677, 6.13613439725025295932208442071, 6.68526356117287462089483685564, 7.51897958150786303110733817438, 7.67713971542231896259485145227, 8.585083425982287523215195122926, 8.823461997305896704784924605521, 9.743408199867838337470935935711, 9.819277877016164792075999364668, 10.31116028768085837893689030303, 10.34648853603495862491516881824, 11.28381770311072609603543422432, 11.85802598157211148736903372086
