| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s − 5·9-s − 3·11-s − 12-s + 16-s − 11·17-s + 5·18-s + 2·19-s + 3·22-s + 24-s − 3·25-s + 8·27-s − 32-s + 3·33-s + 11·34-s − 5·36-s − 2·38-s + 4·41-s + 8·43-s − 3·44-s − 48-s + 2·49-s + 3·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s − 5/3·9-s − 0.904·11-s − 0.288·12-s + 1/4·16-s − 2.66·17-s + 1.17·18-s + 0.458·19-s + 0.639·22-s + 0.204·24-s − 3/5·25-s + 1.53·27-s − 0.176·32-s + 0.522·33-s + 1.88·34-s − 5/6·36-s − 0.324·38-s + 0.624·41-s + 1.21·43-s − 0.452·44-s − 0.144·48-s + 2/7·49-s + 0.424·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 5 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
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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
−10.99255591463655956447942111968, −10.77241963779676766584964161651, −9.977223435935306789670471945603, −9.172805294132700912028932457887, −8.867754255604581457250399746270, −8.336548900780686261398415261268, −7.67427423815294200811839304347, −7.01590572476655019257473150693, −6.19795651628771279823235454090, −5.88050874114616245495556824509, −5.08081251248370835103842521199, −4.28823755499729980221171185102, −2.94424027266467909567067752249, −2.30908705856962981632253808156, 0,
2.30908705856962981632253808156, 2.94424027266467909567067752249, 4.28823755499729980221171185102, 5.08081251248370835103842521199, 5.88050874114616245495556824509, 6.19795651628771279823235454090, 7.01590572476655019257473150693, 7.67427423815294200811839304347, 8.336548900780686261398415261268, 8.867754255604581457250399746270, 9.172805294132700912028932457887, 9.977223435935306789670471945603, 10.77241963779676766584964161651, 10.99255591463655956447942111968
