| L(s) = 1 | − 2·2-s − 3-s + 3·4-s + 2·6-s − 7-s − 4·8-s + 9-s − 3·12-s + 2·14-s + 5·16-s − 2·18-s + 19-s + 21-s + 4·24-s − 25-s − 27-s − 3·28-s − 3·29-s − 6·32-s + 3·36-s − 2·38-s − 7·41-s − 2·42-s + 21·43-s − 5·48-s + 7·49-s + 2·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 3/2·4-s + 0.816·6-s − 0.377·7-s − 1.41·8-s + 1/3·9-s − 0.866·12-s + 0.534·14-s + 5/4·16-s − 0.471·18-s + 0.229·19-s + 0.218·21-s + 0.816·24-s − 1/5·25-s − 0.192·27-s − 0.566·28-s − 0.557·29-s − 1.06·32-s + 1/2·36-s − 0.324·38-s − 1.09·41-s − 0.308·42-s + 3.20·43-s − 0.721·48-s + 49-s + 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 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
−7.971928342399811309163318807084, −7.62868107527512125816999512601, −7.24220719551039278709773228254, −6.99865747244675636948707533342, −6.16950388608483140401582108277, −6.01587866574363804000610737608, −5.63566384072871502776791896481, −4.88563645686008619088412658614, −4.29669999870322491857344366053, −3.71427870031577119124347696256, −2.99968482472843462267367609241, −2.46791734516640255667590875092, −1.69488119140584468940391369752, −0.970526032334489028665696583589, 0,
0.970526032334489028665696583589, 1.69488119140584468940391369752, 2.46791734516640255667590875092, 2.99968482472843462267367609241, 3.71427870031577119124347696256, 4.29669999870322491857344366053, 4.88563645686008619088412658614, 5.63566384072871502776791896481, 6.01587866574363804000610737608, 6.16950388608483140401582108277, 6.99865747244675636948707533342, 7.24220719551039278709773228254, 7.62868107527512125816999512601, 7.971928342399811309163318807084
