| L(s) = 1 | − 12·5-s − 40·13-s − 16·17-s − 142·25-s + 92·29-s + 328·37-s − 624·41-s − 238·49-s − 532·53-s + 264·61-s + 480·65-s + 492·73-s + 192·85-s − 2.78e3·89-s − 604·97-s − 2.93e3·101-s + 3.12e3·109-s + 3.10e3·113-s − 870·121-s + 3.63e3·125-s + 127-s + 131-s + 137-s + 139-s − 1.10e3·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.07·5-s − 0.853·13-s − 0.228·17-s − 1.13·25-s + 0.589·29-s + 1.45·37-s − 2.37·41-s − 0.693·49-s − 1.37·53-s + 0.554·61-s + 0.915·65-s + 0.788·73-s + 0.245·85-s − 3.31·89-s − 0.632·97-s − 2.88·101-s + 2.74·109-s + 2.58·113-s − 0.653·121-s + 2.60·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 0.632·145-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1871032804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1871032804\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 34 p T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 870 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 8 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 6550 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4338 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 46 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 59134 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 164 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 312 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 20186 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 178974 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 266 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 346246 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 132 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 343478 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 257070 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 246 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 931870 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 195606 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1392 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 302 T + p^{3} T^{2} )^{2} \) |
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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
−8.674928589465623250389111698643, −8.368306148613313031431556298209, −8.074641098795012974045236367918, −7.77884339656101187089399900210, −7.29660493386083887982561372464, −7.06506671910476541982437116512, −6.40149049520788482817000551804, −6.37866287016735266993870994060, −5.57463690623759405273983903736, −5.34760665527239682916050832435, −4.64667176483337306434323193170, −4.57937812640227584449594731361, −3.88336965730274095748971935383, −3.73470580477882442327633158365, −2.87690499386731537393282823031, −2.86741936916663348175166069266, −1.94518721903970911902948190406, −1.63370186619493882920138533737, −0.795229420565376285503701111966, −0.10829014482775284369731266992,
0.10829014482775284369731266992, 0.795229420565376285503701111966, 1.63370186619493882920138533737, 1.94518721903970911902948190406, 2.86741936916663348175166069266, 2.87690499386731537393282823031, 3.73470580477882442327633158365, 3.88336965730274095748971935383, 4.57937812640227584449594731361, 4.64667176483337306434323193170, 5.34760665527239682916050832435, 5.57463690623759405273983903736, 6.37866287016735266993870994060, 6.40149049520788482817000551804, 7.06506671910476541982437116512, 7.29660493386083887982561372464, 7.77884339656101187089399900210, 8.074641098795012974045236367918, 8.368306148613313031431556298209, 8.674928589465623250389111698643
