| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s − 2·9-s − 3·11-s − 12-s + 16-s + 2·18-s − 3·19-s + 3·22-s + 24-s + 7·25-s + 5·27-s − 32-s + 3·33-s − 2·36-s + 3·38-s − 15·41-s + 14·43-s − 3·44-s − 48-s + 6·49-s − 7·50-s − 5·54-s + 3·57-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s − 2/3·9-s − 0.904·11-s − 0.288·12-s + 1/4·16-s + 0.471·18-s − 0.688·19-s + 0.639·22-s + 0.204·24-s + 7/5·25-s + 0.962·27-s − 0.176·32-s + 0.522·33-s − 1/3·36-s + 0.486·38-s − 2.34·41-s + 2.13·43-s − 0.452·44-s − 0.144·48-s + 6/7·49-s − 0.989·50-s − 0.680·54-s + 0.397·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 43 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 107 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
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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
−8.599569373055739099398430093325, −8.243395735717040315208190823106, −7.57053868656842822299324657103, −7.17061600949874038740815379553, −6.74297579027304684065669752546, −6.11389130747304760389727063730, −5.82167479893925239775852515335, −5.16063838149344382142514436928, −4.84051543331438606923344010770, −4.09358748581592722649542913924, −3.27351388306579564969765149734, −2.72888652964056413613858980759, −2.15497184161703396786780118670, −1.05407845064146052673788820632, 0,
1.05407845064146052673788820632, 2.15497184161703396786780118670, 2.72888652964056413613858980759, 3.27351388306579564969765149734, 4.09358748581592722649542913924, 4.84051543331438606923344010770, 5.16063838149344382142514436928, 5.82167479893925239775852515335, 6.11389130747304760389727063730, 6.74297579027304684065669752546, 7.17061600949874038740815379553, 7.57053868656842822299324657103, 8.243395735717040315208190823106, 8.599569373055739099398430093325
