L(s) = 1 | − 2·4-s − 2·7-s − 5·13-s + 4·16-s − 2·19-s − 25-s + 4·28-s − 8·31-s − 5·37-s − 20·43-s + 7·49-s + 10·52-s − 5·61-s − 8·64-s − 2·67-s + 73-s + 4·76-s + 10·79-s + 10·91-s + 10·97-s + 2·100-s + 101-s + 103-s + 107-s + 109-s − 8·112-s + 113-s + ⋯ |
L(s) = 1 | − 4-s − 0.755·7-s − 1.38·13-s + 16-s − 0.458·19-s − 1/5·25-s + 0.755·28-s − 1.43·31-s − 0.821·37-s − 3.04·43-s + 49-s + 1.38·52-s − 0.640·61-s − 64-s − 0.244·67-s + 0.117·73-s + 0.458·76-s + 1.12·79-s + 1.04·91-s + 1.01·97-s + 1/5·100-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s − 0.755·112-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−15.5661348379, −14.9978425101, −14.8618552521, −14.2503916444, −13.7566908660, −13.3063109520, −12.8913822048, −12.4454649925, −12.0211228594, −11.5204763615, −10.6345411192, −10.2908491834, −9.78188532983, −9.37050253492, −8.85577608372, −8.32250088323, −7.66007083372, −7.10233129189, −6.52504595094, −5.73730543418, −5.11207898163, −4.64259559899, −3.72482723315, −3.18716773167, −1.98532177689, 0,
1.98532177689, 3.18716773167, 3.72482723315, 4.64259559899, 5.11207898163, 5.73730543418, 6.52504595094, 7.10233129189, 7.66007083372, 8.32250088323, 8.85577608372, 9.37050253492, 9.78188532983, 10.2908491834, 10.6345411192, 11.5204763615, 12.0211228594, 12.4454649925, 12.8913822048, 13.3063109520, 13.7566908660, 14.2503916444, 14.8618552521, 14.9978425101, 15.5661348379