| L(s) = 1 | − 4-s − 2·5-s − 9-s + 4·11-s + 16-s − 12·19-s + 2·20-s − 25-s − 12·29-s + 4·31-s + 36-s − 4·41-s − 4·44-s + 2·45-s − 8·55-s − 16·59-s + 20·61-s − 64-s − 12·71-s + 12·76-s + 24·79-s − 2·80-s + 81-s − 20·89-s + 24·95-s − 4·99-s + 100-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.894·5-s − 1/3·9-s + 1.20·11-s + 1/4·16-s − 2.75·19-s + 0.447·20-s − 1/5·25-s − 2.22·29-s + 0.718·31-s + 1/6·36-s − 0.624·41-s − 0.603·44-s + 0.298·45-s − 1.07·55-s − 2.08·59-s + 2.56·61-s − 1/8·64-s − 1.42·71-s + 1.37·76-s + 2.70·79-s − 0.223·80-s + 1/9·81-s − 2.11·89-s + 2.46·95-s − 0.402·99-s + 1/10·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4528072374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4528072374\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.817672393704331028914981915898, −9.243029968623922934561804248419, −8.761751162750568859363025610708, −8.496411945930032903184952838663, −8.345444419641659704237362257480, −7.64889363709902841040397686416, −7.43150341820798187515509074942, −6.83557115096762224016837734688, −6.38505691649570084556754924588, −6.16296526541492960489266925645, −5.62634420081571443108872283508, −5.02980922003753405234051350815, −4.54127714516856148943948889454, −4.10231931860121892725461323508, −3.79205522420904522971850621723, −3.50732513391559170873267986052, −2.60102075713141769202822453220, −2.03881279619832219134759600815, −1.40952101175636141981813424684, −0.27455764657450222375701516397,
0.27455764657450222375701516397, 1.40952101175636141981813424684, 2.03881279619832219134759600815, 2.60102075713141769202822453220, 3.50732513391559170873267986052, 3.79205522420904522971850621723, 4.10231931860121892725461323508, 4.54127714516856148943948889454, 5.02980922003753405234051350815, 5.62634420081571443108872283508, 6.16296526541492960489266925645, 6.38505691649570084556754924588, 6.83557115096762224016837734688, 7.43150341820798187515509074942, 7.64889363709902841040397686416, 8.345444419641659704237362257480, 8.496411945930032903184952838663, 8.761751162750568859363025610708, 9.243029968623922934561804248419, 9.817672393704331028914981915898
