L(s) = 1 | − 3-s − 10·7-s + 9-s + 4·13-s − 4·16-s − 2·19-s + 10·21-s − 25-s − 27-s − 12·31-s − 4·39-s − 2·43-s + 4·48-s + 61·49-s + 2·57-s − 2·61-s − 10·63-s + 16·67-s − 22·73-s + 75-s + 32·79-s + 81-s − 40·91-s + 12·93-s − 20·97-s − 4·103-s + 8·109-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 3.77·7-s + 1/3·9-s + 1.10·13-s − 16-s − 0.458·19-s + 2.18·21-s − 1/5·25-s − 0.192·27-s − 2.15·31-s − 0.640·39-s − 0.304·43-s + 0.577·48-s + 61/7·49-s + 0.264·57-s − 0.256·61-s − 1.25·63-s + 1.95·67-s − 2.57·73-s + 0.115·75-s + 3.60·79-s + 1/9·81-s − 4.19·91-s + 1.24·93-s − 2.03·97-s − 0.394·103-s + 0.766·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9747 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9747 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−11.00179000913096952209865162760, −10.84385583714429238920836096770, −9.914070570189995801171942379165, −9.769325574936890193507770039971, −9.028755790922139382896168795846, −8.836427381447004052087018681826, −7.47848778675296589128891353641, −6.85073886350288365363080648917, −6.41173993136661837405396202601, −6.12241751609850681792173625996, −5.34668146252658900228619106104, −3.81591641511052084615077615666, −3.67840796781373175574076394185, −2.61687998381550726785939862479, 0,
2.61687998381550726785939862479, 3.67840796781373175574076394185, 3.81591641511052084615077615666, 5.34668146252658900228619106104, 6.12241751609850681792173625996, 6.41173993136661837405396202601, 6.85073886350288365363080648917, 7.47848778675296589128891353641, 8.836427381447004052087018681826, 9.028755790922139382896168795846, 9.769325574936890193507770039971, 9.914070570189995801171942379165, 10.84385583714429238920836096770, 11.00179000913096952209865162760