L(s) = 1 | + 48·13-s + 48·25-s + 140·37-s − 98·49-s − 44·61-s − 192·73-s − 288·97-s − 240·109-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.39e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
L(s) = 1 | + 3.69·13-s + 1.91·25-s + 3.78·37-s − 2·49-s − 0.721·61-s − 2.63·73-s − 2.96·97-s − 2.20·109-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 + 8.22·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.862792754\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.862792754\) |
\(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 48 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 480 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 1680 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 1440 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 5040 T^{2} + p^{4} T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 96 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12480 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 144 T + p^{2} T^{2} )^{2} \) |
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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
−11.42697745112321182878120258018, −11.40455288084448529170697657602, −10.86894351261976665100392060064, −10.73444681161266803411605178172, −9.905915598574959002864075178796, −9.430786796458750080347435256997, −8.761097516980137156698371596482, −8.647113939972459696625117725755, −8.034030674122440330313574295014, −7.64792360165116751020659223077, −6.53885104430615464515509703611, −6.53352508491681543213215877957, −5.92560697305733138129199407580, −5.44718075636809267901643654807, −4.32269421092743956461936916476, −4.24301766966859931064342635227, −3.23990056100399394213424254890, −2.88628165958439073310052056447, −1.44704203259757710934521997889, −1.00765691716622985797871660799,
1.00765691716622985797871660799, 1.44704203259757710934521997889, 2.88628165958439073310052056447, 3.23990056100399394213424254890, 4.24301766966859931064342635227, 4.32269421092743956461936916476, 5.44718075636809267901643654807, 5.92560697305733138129199407580, 6.53352508491681543213215877957, 6.53885104430615464515509703611, 7.64792360165116751020659223077, 8.034030674122440330313574295014, 8.647113939972459696625117725755, 8.761097516980137156698371596482, 9.430786796458750080347435256997, 9.905915598574959002864075178796, 10.73444681161266803411605178172, 10.86894351261976665100392060064, 11.40455288084448529170697657602, 11.42697745112321182878120258018