| L(s) = 1 | − 2·3-s − 7-s + 9-s + 4·13-s + 4·19-s + 2·21-s + 6·25-s + 4·27-s − 8·31-s − 4·37-s − 8·39-s + 49-s − 8·57-s − 4·61-s − 63-s − 16·67-s + 16·73-s − 12·75-s − 16·79-s − 11·81-s − 4·91-s + 16·93-s − 24·97-s − 24·103-s − 4·109-s + 8·111-s + 4·117-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.10·13-s + 0.917·19-s + 0.436·21-s + 6/5·25-s + 0.769·27-s − 1.43·31-s − 0.657·37-s − 1.28·39-s + 1/7·49-s − 1.05·57-s − 0.512·61-s − 0.125·63-s − 1.95·67-s + 1.87·73-s − 1.38·75-s − 1.80·79-s − 1.22·81-s − 0.419·91-s + 1.65·93-s − 2.43·97-s − 2.36·103-s − 0.383·109-s + 0.759·111-s + 0.369·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.088389702199429563164563679829, −7.40619201544090395217663098448, −7.04659213070794210454373540616, −6.64178816595729548798645800119, −6.22150256805029902828972726936, −5.71048407119543373439551017803, −5.36119136560208897119945760211, −5.04780507640820270646940449869, −4.27051574386605810601973703544, −3.87443310065837143952413124388, −3.12542015073637113846082429317, −2.81887205063555345627861372389, −1.65559074694283709969529680898, −1.07814534596256427072453324339, 0,
1.07814534596256427072453324339, 1.65559074694283709969529680898, 2.81887205063555345627861372389, 3.12542015073637113846082429317, 3.87443310065837143952413124388, 4.27051574386605810601973703544, 5.04780507640820270646940449869, 5.36119136560208897119945760211, 5.71048407119543373439551017803, 6.22150256805029902828972726936, 6.64178816595729548798645800119, 7.04659213070794210454373540616, 7.40619201544090395217663098448, 8.088389702199429563164563679829
