| L(s) = 1 | − 4·2-s + 8·4-s − 8·8-s + 3·9-s + 2·11-s − 4·16-s − 12·18-s − 8·22-s − 8·23-s + 25-s − 2·29-s + 32·32-s + 24·36-s − 12·43-s + 16·44-s + 32·46-s − 4·50-s − 20·53-s + 8·58-s − 64·64-s − 28·67-s − 16·71-s − 24·72-s − 2·79-s + 48·86-s − 16·88-s − 64·92-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4·4-s − 2.82·8-s + 9-s + 0.603·11-s − 16-s − 2.82·18-s − 1.70·22-s − 1.66·23-s + 1/5·25-s − 0.371·29-s + 5.65·32-s + 4·36-s − 1.82·43-s + 2.41·44-s + 4.71·46-s − 0.565·50-s − 2.74·53-s + 1.05·58-s − 8·64-s − 3.42·67-s − 1.89·71-s − 2.82·72-s − 0.225·79-s + 5.17·86-s − 1.70·88-s − 6.67·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 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
−9.674071857485304380380573397765, −9.154994581895650846359851030198, −8.895041085769743028052283031023, −8.210184381994137443519230420468, −7.79509008882088804636414623325, −7.56848203557801886905576002740, −6.67134380816033742861497166759, −6.66812515626338690414111791742, −5.69074574951383439724683067976, −4.49258853804675338518966926948, −4.33023383010768085745155570778, −3.07565995342284756040604408072, −1.70898778880961935771596537039, −1.56775892122303114558742737725, 0,
1.56775892122303114558742737725, 1.70898778880961935771596537039, 3.07565995342284756040604408072, 4.33023383010768085745155570778, 4.49258853804675338518966926948, 5.69074574951383439724683067976, 6.66812515626338690414111791742, 6.67134380816033742861497166759, 7.56848203557801886905576002740, 7.79509008882088804636414623325, 8.210184381994137443519230420468, 8.895041085769743028052283031023, 9.154994581895650846359851030198, 9.674071857485304380380573397765
