| L(s) = 1 | − 32·17-s − 14·25-s − 160·41-s + 98·49-s − 220·73-s + 320·89-s − 260·97-s + 448·113-s − 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 238·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 1.88·17-s − 0.559·25-s − 3.90·41-s + 2·49-s − 3.01·73-s + 3.59·89-s − 2.68·97-s + 3.96·113-s − 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.40·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1838476365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1838476365\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )( 1 + 6 T + p^{2} T^{2} ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )( 1 + 24 T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )( 1 + 42 T + p^{2} T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )( 1 + 24 T + p^{2} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 80 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 90 T + p^{2} T^{2} )( 1 + 90 T + p^{2} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 120 T + p^{2} T^{2} )( 1 + 120 T + p^{2} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 110 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 160 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 130 T + p^{2} T^{2} )^{2} \) |
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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
−9.232209136851508745226665343860, −8.663925471996682732216225294959, −8.410717922423940163256048948958, −7.85270983954925281257458878700, −7.46170853131590429054819953335, −7.00292305791717329107484390345, −6.73710319460723332686557801606, −6.40632977768699523168355530942, −5.87291774803100614173443823406, −5.50588378388715542281210878335, −5.01722402598296722380591871320, −4.52149573289662922536254066216, −4.36135707365332285886039434454, −3.52158219445314296362992092321, −3.48727403363898510937007883629, −2.65013416968999020829653281068, −2.20624476668426273428652406136, −1.77237310731648065597035701029, −1.12192597085330551879615934889, −0.10680510751916947564980557105,
0.10680510751916947564980557105, 1.12192597085330551879615934889, 1.77237310731648065597035701029, 2.20624476668426273428652406136, 2.65013416968999020829653281068, 3.48727403363898510937007883629, 3.52158219445314296362992092321, 4.36135707365332285886039434454, 4.52149573289662922536254066216, 5.01722402598296722380591871320, 5.50588378388715542281210878335, 5.87291774803100614173443823406, 6.40632977768699523168355530942, 6.73710319460723332686557801606, 7.00292305791717329107484390345, 7.46170853131590429054819953335, 7.85270983954925281257458878700, 8.410717922423940163256048948958, 8.663925471996682732216225294959, 9.232209136851508745226665343860
