| L(s) = 1 | − 2·3-s − 4-s + 2·5-s + 3·9-s + 11-s + 2·12-s − 4·15-s + 16-s − 2·20-s + 2·23-s − 6·25-s − 4·27-s − 8·31-s − 2·33-s − 3·36-s + 4·37-s − 44-s + 6·45-s − 2·47-s − 2·48-s + 49-s − 2·53-s + 2·55-s + 4·60-s − 64-s − 4·69-s + 10·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 0.894·5-s + 9-s + 0.301·11-s + 0.577·12-s − 1.03·15-s + 1/4·16-s − 0.447·20-s + 0.417·23-s − 6/5·25-s − 0.769·27-s − 1.43·31-s − 0.348·33-s − 1/2·36-s + 0.657·37-s − 0.150·44-s + 0.894·45-s − 0.291·47-s − 0.288·48-s + 1/7·49-s − 0.274·53-s + 0.269·55-s + 0.516·60-s − 1/8·64-s − 0.481·69-s + 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2347884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2347884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−7.41770998509546734037712728327, −6.81539110196714991218053395773, −6.66141965171668532695097678701, −6.00887630839035476264421082560, −5.79295625950230208692917141974, −5.41632630190641561348112455249, −4.99232032691374663214855023871, −4.55759299848791033942009421313, −3.91165220490796163542037045395, −3.69912265401621314150175657358, −2.88447614349323662960812654781, −2.11455881782595400347484153694, −1.67832160592333745510581620712, −0.927973759361932525745661905814, 0,
0.927973759361932525745661905814, 1.67832160592333745510581620712, 2.11455881782595400347484153694, 2.88447614349323662960812654781, 3.69912265401621314150175657358, 3.91165220490796163542037045395, 4.55759299848791033942009421313, 4.99232032691374663214855023871, 5.41632630190641561348112455249, 5.79295625950230208692917141974, 6.00887630839035476264421082560, 6.66141965171668532695097678701, 6.81539110196714991218053395773, 7.41770998509546734037712728327
