| L(s) = 1 | − 384·17-s + 116·25-s + 1.15e3·41-s − 1.34e3·49-s − 2.15e3·73-s − 5.37e3·89-s − 2.36e3·97-s + 768·113-s + 716·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.92e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 5.47·17-s + 0.927·25-s + 4.38·41-s − 3.93·49-s − 3.45·73-s − 6.40·89-s − 2.47·97-s + 0.639·113-s + 0.537·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 3.60·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5554546771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5554546771\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 674 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 358 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 3962 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 96 T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 12118 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 12046 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 48586 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 17810 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 12554 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 288 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 135910 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 99554 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 265306 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 180358 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 116326 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 117578 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 70610 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 538 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 30094 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 956950 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 1344 T + p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 590 T + p^{3} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.09932203002597565994470164810, −7.02991556412982638618126072465, −6.93297668198878902858378799583, −6.57421767281394597707947129505, −6.45688701946566511894266438976, −6.11639578024126330290441820489, −5.87630813846473721930957690686, −5.76969852119645143867107601574, −5.26892789920994912137480514537, −5.09457672981485053473622289358, −4.57288351966548682677435643443, −4.33797200267635410905716941495, −4.33429475238747031501053304008, −4.26322150389409097728763187995, −4.10152816305759769137667084923, −3.20450728824967372718284750350, −2.95417428465133995472283705268, −2.94771208442251610340594274755, −2.35807579429580258950831657477, −2.29572692098964820748320495240, −1.92878372782111158257271379353, −1.38799773224146188372363016184, −1.26034651133895404860356668779, −0.38825598537951438965242996898, −0.17231599488565185040479764797,
0.17231599488565185040479764797, 0.38825598537951438965242996898, 1.26034651133895404860356668779, 1.38799773224146188372363016184, 1.92878372782111158257271379353, 2.29572692098964820748320495240, 2.35807579429580258950831657477, 2.94771208442251610340594274755, 2.95417428465133995472283705268, 3.20450728824967372718284750350, 4.10152816305759769137667084923, 4.26322150389409097728763187995, 4.33429475238747031501053304008, 4.33797200267635410905716941495, 4.57288351966548682677435643443, 5.09457672981485053473622289358, 5.26892789920994912137480514537, 5.76969852119645143867107601574, 5.87630813846473721930957690686, 6.11639578024126330290441820489, 6.45688701946566511894266438976, 6.57421767281394597707947129505, 6.93297668198878902858378799583, 7.02991556412982638618126072465, 7.09932203002597565994470164810
