| L(s) = 1 | − 3-s − 2·4-s − 2·9-s − 2·11-s + 2·12-s + 4·16-s − 4·17-s − 8·19-s − 2·25-s + 2·27-s + 2·33-s + 4·36-s − 6·43-s + 4·44-s − 4·48-s − 2·49-s + 4·51-s + 8·57-s − 2·59-s − 8·64-s − 12·67-s + 8·68-s + 5·73-s + 2·75-s + 16·76-s + 7·81-s + 2·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 2/3·9-s − 0.603·11-s + 0.577·12-s + 16-s − 0.970·17-s − 1.83·19-s − 2/5·25-s + 0.384·27-s + 0.348·33-s + 2/3·36-s − 0.914·43-s + 0.603·44-s − 0.577·48-s − 2/7·49-s + 0.560·51-s + 1.05·57-s − 0.260·59-s − 64-s − 1.46·67-s + 0.970·68-s + 0.585·73-s + 0.230·75-s + 1.83·76-s + 7/9·81-s + 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−10.70483665499183740328680292184, −10.59372501985034855221719026342, −9.898519199433535909263524895787, −9.070034127943175221378657054694, −8.845890212499120528013656082094, −8.147206201352259471051285188036, −7.72931680764636423645282073838, −6.66094801634365823017137659041, −6.20218989411660605978029505394, −5.52847886146026909243450505195, −4.83708225911762336640984424202, −4.31190301832617759674310954283, −3.38045649380199875812514092211, −2.19471650371950094844974434167, 0,
2.19471650371950094844974434167, 3.38045649380199875812514092211, 4.31190301832617759674310954283, 4.83708225911762336640984424202, 5.52847886146026909243450505195, 6.20218989411660605978029505394, 6.66094801634365823017137659041, 7.72931680764636423645282073838, 8.147206201352259471051285188036, 8.845890212499120528013656082094, 9.070034127943175221378657054694, 9.898519199433535909263524895787, 10.59372501985034855221719026342, 10.70483665499183740328680292184
