| L(s) = 1 | − 2·3-s + 4-s + 4·5-s + 3·9-s + 11-s − 2·12-s − 8·15-s + 16-s + 4·20-s + 8·23-s + 2·25-s − 4·27-s + 16·31-s − 2·33-s + 3·36-s + 12·37-s + 44-s + 12·45-s + 8·47-s − 2·48-s + 49-s + 20·53-s + 4·55-s + 8·59-s − 8·60-s + 64-s − 8·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 1.78·5-s + 9-s + 0.301·11-s − 0.577·12-s − 2.06·15-s + 1/4·16-s + 0.894·20-s + 1.66·23-s + 2/5·25-s − 0.769·27-s + 2.87·31-s − 0.348·33-s + 1/2·36-s + 1.97·37-s + 0.150·44-s + 1.78·45-s + 1.16·47-s − 0.288·48-s + 1/7·49-s + 2.74·53-s + 0.539·55-s + 1.04·59-s − 1.03·60-s + 1/8·64-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2347884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2347884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.312210756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.312210756\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.34608818311813371377464448078, −7.18348937533447218632490891225, −6.72668324855219479594237702896, −6.21868608096758870562967590194, −5.92489513820890387415408647541, −5.83815977597462037088554178569, −5.20978693475542734341677894398, −4.83426317009020569023368578340, −4.27546340147020629083452713682, −3.87226536522948976697296552112, −2.77059081857105425612062855601, −2.65895736335338182549695498478, −2.03409091621658650070101527627, −1.14483701940607571217981775775, −0.946521678284674219466217873830,
0.946521678284674219466217873830, 1.14483701940607571217981775775, 2.03409091621658650070101527627, 2.65895736335338182549695498478, 2.77059081857105425612062855601, 3.87226536522948976697296552112, 4.27546340147020629083452713682, 4.83426317009020569023368578340, 5.20978693475542734341677894398, 5.83815977597462037088554178569, 5.92489513820890387415408647541, 6.21868608096758870562967590194, 6.72668324855219479594237702896, 7.18348937533447218632490891225, 7.34608818311813371377464448078
