| L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s − 3·8-s − 2·9-s + 10-s − 12-s + 6·13-s + 15-s − 16-s − 2·18-s − 20-s − 3·24-s + 25-s + 6·26-s − 5·27-s + 30-s − 17·31-s + 5·32-s + 2·36-s − 6·37-s + 6·39-s − 3·40-s − 12·41-s + 12·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.06·8-s − 2/3·9-s + 0.316·10-s − 0.288·12-s + 1.66·13-s + 0.258·15-s − 1/4·16-s − 0.471·18-s − 0.223·20-s − 0.612·24-s + 1/5·25-s + 1.17·26-s − 0.962·27-s + 0.182·30-s − 3.05·31-s + 0.883·32-s + 1/3·36-s − 0.986·37-s + 0.960·39-s − 0.474·40-s − 1.87·41-s + 1.82·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$ | \( 1 - T \) |
| good | 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 149 T^{2} + p^{2} T^{4} \) |
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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
−8.404892992106707770292934217184, −7.73426381839069259576548229253, −7.23713669171286017664751529072, −6.69402496526277030300863872500, −6.06687461098879686613562001077, −5.86823014230672516057217455980, −5.30625443985018247491656024388, −5.06402787071068493214278367179, −4.19126219943191400695945427909, −3.60854845966419712047397884569, −3.55382098118598749748085893604, −2.86125002678782704502545145282, −2.07346165375381666620239943963, −1.39691262973298443502724962073, 0,
1.39691262973298443502724962073, 2.07346165375381666620239943963, 2.86125002678782704502545145282, 3.55382098118598749748085893604, 3.60854845966419712047397884569, 4.19126219943191400695945427909, 5.06402787071068493214278367179, 5.30625443985018247491656024388, 5.86823014230672516057217455980, 6.06687461098879686613562001077, 6.69402496526277030300863872500, 7.23713669171286017664751529072, 7.73426381839069259576548229253, 8.404892992106707770292934217184
