| L(s) = 1 | − 24·11-s + 16-s − 16·31-s + 12·41-s − 32·61-s − 24·101-s + 268·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 24·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 7.23·11-s + 1/4·16-s − 2.87·31-s + 1.87·41-s − 4.09·61-s − 2.38·101-s + 24.3·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s − 1.80·176-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3327642688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3327642688\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{4} + T^{8} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( 1 + 142 T^{4} - 8397 T^{8} + 142 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 + 958 T^{4} + 637923 T^{8} + 958 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 10 T^{2} - 741 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 7 T + p T^{2} )^{4}( 1 + 11 T + p T^{2} )^{4} \) |
| 37 | \( ( 1 - 2062 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 3 T + 44 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 3214 T^{4} + p^{4} T^{8} )( 1 + 3191 T^{4} + p^{4} T^{8} ) \) |
| 47 | \( 1 + 1054 T^{4} - 3768765 T^{8} + 1054 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 718 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 115 T^{2} + 9744 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 + 5497 T^{4} + 10065888 T^{8} + 5497 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 10657 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 38 T^{2} - 4797 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 6553 T^{4} - 4516512 T^{8} + 6553 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 18814 T^{4} + p^{4} T^{8} )( 1 + 9743 T^{4} + p^{4} T^{8} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.03315511475874063734508270302, −3.89967920854682018397757669063, −3.84494932431761278404131228001, −3.66075578043579037988752687398, −3.44301172818004495886731855967, −3.41461576166585817468218949179, −3.37102128508124179314405231337, −3.09186167636952181843603397234, −2.85970825819182466366197313576, −2.75766177741297710221695946329, −2.72844858829517088556094517289, −2.60254083350084254682804552394, −2.49816346797713282965529281664, −2.45823413536895443069158487152, −2.45519470471167613237535167824, −2.17184774204511474164545603884, −1.76226485947900671359584359300, −1.73160998500716616965290274855, −1.56374990951191085579934294073, −1.39200427565062731752158788716, −1.37841394837615382744794933492, −0.66273122598204034820750098419, −0.45574769133996219019952003101, −0.39549994904944847830084005680, −0.13574206832900278375566755136,
0.13574206832900278375566755136, 0.39549994904944847830084005680, 0.45574769133996219019952003101, 0.66273122598204034820750098419, 1.37841394837615382744794933492, 1.39200427565062731752158788716, 1.56374990951191085579934294073, 1.73160998500716616965290274855, 1.76226485947900671359584359300, 2.17184774204511474164545603884, 2.45519470471167613237535167824, 2.45823413536895443069158487152, 2.49816346797713282965529281664, 2.60254083350084254682804552394, 2.72844858829517088556094517289, 2.75766177741297710221695946329, 2.85970825819182466366197313576, 3.09186167636952181843603397234, 3.37102128508124179314405231337, 3.41461576166585817468218949179, 3.44301172818004495886731855967, 3.66075578043579037988752687398, 3.84494932431761278404131228001, 3.89967920854682018397757669063, 4.03315511475874063734508270302
Plot not available for L-functions of degree greater than 10.
