| L(s) = 1 | + 3-s + 4-s + 2·5-s − 7-s + 9-s + 12-s + 2·15-s + 16-s + 4·17-s + 2·20-s − 21-s + 3·25-s + 27-s − 28-s − 2·35-s + 36-s + 12·37-s − 12·41-s − 8·43-s + 2·45-s + 48-s + 49-s + 4·51-s + 24·59-s + 2·60-s − 63-s + 64-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.288·12-s + 0.516·15-s + 1/4·16-s + 0.970·17-s + 0.447·20-s − 0.218·21-s + 3/5·25-s + 0.192·27-s − 0.188·28-s − 0.338·35-s + 1/6·36-s + 1.97·37-s − 1.87·41-s − 1.21·43-s + 0.298·45-s + 0.144·48-s + 1/7·49-s + 0.560·51-s + 3.12·59-s + 0.258·60-s − 0.125·63-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 926100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 926100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.674923837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.674923837\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.185634457079877915713969006222, −7.68131261207395042033915827358, −7.31970008287706573052341981271, −6.81537318886956844914397666067, −6.40973998919568114415810052240, −5.97975998469911468351052676550, −5.59102503422980904290428466069, −4.95725936124254114017607791343, −4.62242475448465894529250370260, −3.67516322565973252877181398017, −3.44893990823261081236809071688, −2.79601598717148031870127332799, −2.27051973415844153304852765225, −1.69559251292467624738091785885, −0.897247117961030373363801846777,
0.897247117961030373363801846777, 1.69559251292467624738091785885, 2.27051973415844153304852765225, 2.79601598717148031870127332799, 3.44893990823261081236809071688, 3.67516322565973252877181398017, 4.62242475448465894529250370260, 4.95725936124254114017607791343, 5.59102503422980904290428466069, 5.97975998469911468351052676550, 6.40973998919568114415810052240, 6.81537318886956844914397666067, 7.31970008287706573052341981271, 7.68131261207395042033915827358, 8.185634457079877915713969006222
