L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 7-s − 3·8-s + 9-s − 8·11-s − 12-s − 4·13-s − 14-s − 16-s + 12·17-s + 18-s + 8·19-s − 21-s − 8·22-s − 3·24-s − 6·25-s − 4·26-s + 27-s + 28-s + 4·29-s + 5·32-s − 8·33-s + 12·34-s − 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s − 2.41·11-s − 0.288·12-s − 1.10·13-s − 0.267·14-s − 1/4·16-s + 2.91·17-s + 0.235·18-s + 1.83·19-s − 0.218·21-s − 1.70·22-s − 0.612·24-s − 6/5·25-s − 0.784·26-s + 0.192·27-s + 0.188·28-s + 0.742·29-s + 0.883·32-s − 1.39·33-s + 2.05·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$ | \( 1 + T \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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
−8.047316704349055543988739384910, −7.81295430367747747098376616892, −7.41162979563327424538391535036, −7.08463284614449031614302730659, −5.94607665445287604157300008429, −5.84366683169447599172191319438, −5.25233061805071063226919136648, −5.02709416296405066539370067618, −4.56778712711941505527081631821, −3.63417394911967134530507000276, −3.13295449004887072368969635069, −3.05422074105458389777226041971, −2.36822825472434546359232655114, −1.19651242282215311581594916621, 0,
1.19651242282215311581594916621, 2.36822825472434546359232655114, 3.05422074105458389777226041971, 3.13295449004887072368969635069, 3.63417394911967134530507000276, 4.56778712711941505527081631821, 5.02709416296405066539370067618, 5.25233061805071063226919136648, 5.84366683169447599172191319438, 5.94607665445287604157300008429, 7.08463284614449031614302730659, 7.41162979563327424538391535036, 7.81295430367747747098376616892, 8.047316704349055543988739384910