| L(s) = 1 | − 2-s + 2·3-s − 4-s − 5-s − 2·6-s + 3·8-s + 9-s + 10-s − 2·12-s + 3·13-s − 2·15-s − 16-s − 18-s + 20-s + 6·24-s + 25-s − 3·26-s − 4·27-s + 2·30-s − 5·31-s − 5·32-s − 36-s − 21·37-s + 6·39-s − 3·40-s − 12·41-s − 6·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.447·5-s − 0.816·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.577·12-s + 0.832·13-s − 0.516·15-s − 1/4·16-s − 0.235·18-s + 0.223·20-s + 1.22·24-s + 1/5·25-s − 0.588·26-s − 0.769·27-s + 0.365·30-s − 0.898·31-s − 0.883·32-s − 1/6·36-s − 3.45·37-s + 0.960·39-s − 0.474·40-s − 1.87·41-s − 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 48 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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
−8.699826047687082336822058020289, −8.407925194342252611576426328430, −8.231152763340728022945344952135, −7.47870553299200018880941726522, −7.10715731494937986420727448761, −6.70285091451097027770080096620, −5.83393016259382569689004866228, −5.17580431414177339059990269329, −4.86833177044701056643889722537, −3.81474994039728055277844345254, −3.68885867024609674771432523082, −3.15262185397536829081504012922, −2.04035771938488058529775286099, −1.49891342858877232756243501199, 0,
1.49891342858877232756243501199, 2.04035771938488058529775286099, 3.15262185397536829081504012922, 3.68885867024609674771432523082, 3.81474994039728055277844345254, 4.86833177044701056643889722537, 5.17580431414177339059990269329, 5.83393016259382569689004866228, 6.70285091451097027770080096620, 7.10715731494937986420727448761, 7.47870553299200018880941726522, 8.231152763340728022945344952135, 8.407925194342252611576426328430, 8.699826047687082336822058020289
