| 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 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.588553744868809505827514460350, −7.67446852392043612593360900832, −7.25047783802838427330028450541, −7.05615307295907564528444841462, −6.63683595768206904498528521377, −5.92132610292759546410040486711, −5.49363994574552256101491493631, −4.64476747144253945214734257534, −4.15191135596254117729962430013, −4.13559084050773741974089362056, −3.34465725274578742569214950483, −2.53216049664136959139958902434, −1.86996783063205051384617951968, −1.21485222539328476495336836851, 0,
1.21485222539328476495336836851, 1.86996783063205051384617951968, 2.53216049664136959139958902434, 3.34465725274578742569214950483, 4.13559084050773741974089362056, 4.15191135596254117729962430013, 4.64476747144253945214734257534, 5.49363994574552256101491493631, 5.92132610292759546410040486711, 6.63683595768206904498528521377, 7.05615307295907564528444841462, 7.25047783802838427330028450541, 7.67446852392043612593360900832, 8.588553744868809505827514460350
