| L(s) = 1 | − 272·11-s + 47·16-s − 2.75e3·71-s − 1.45e3·81-s + 4.09e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 1.27e4·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 7.45·11-s + 0.734·16-s − 4.60·71-s − 2·81-s + 30.7·121-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 + 0.000439·173-s − 5.47·176-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 + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.07680812826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07680812826\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 - 47 T^{4} + p^{12} T^{8} \) |
| 3 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 68 T + p^{3} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 220762978 T^{4} + p^{12} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 21222 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 5108772818 T^{4} + p^{12} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 + 3388378898 T^{4} + p^{12} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - 41794002542 T^{4} + p^{12} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 178008750862 T^{4} + p^{12} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 688 T + p^{3} T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 929378 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
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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
−8.640224288404582557307461277710, −8.370610222134724640929864522863, −8.236570732183097282250342914697, −7.991258178551428809923930705927, −7.74555136436794770677100913647, −7.47173950046536976078516747258, −7.46331836359875042521779652762, −7.11679222730311977311243313570, −6.70394791502920607763606604143, −5.97690562865895347125808793817, −5.62633404211620869393324220050, −5.61080965042280451684357782787, −5.59397943403898716129322363575, −5.09685804934577376440588097949, −4.82373809281802209764620022277, −4.48792791222955230410033907439, −4.31984668525214712392942927160, −3.14210041408009617356806572783, −3.07690261543908491846916128499, −3.05164589508112450238436078938, −2.44898171492519458668750014395, −2.30417444750615180976516682607, −1.72179167152561135456301288526, −0.59044095785827505405162335093, −0.089350533331963175299704238643,
0.089350533331963175299704238643, 0.59044095785827505405162335093, 1.72179167152561135456301288526, 2.30417444750615180976516682607, 2.44898171492519458668750014395, 3.05164589508112450238436078938, 3.07690261543908491846916128499, 3.14210041408009617356806572783, 4.31984668525214712392942927160, 4.48792791222955230410033907439, 4.82373809281802209764620022277, 5.09685804934577376440588097949, 5.59397943403898716129322363575, 5.61080965042280451684357782787, 5.62633404211620869393324220050, 5.97690562865895347125808793817, 6.70394791502920607763606604143, 7.11679222730311977311243313570, 7.46331836359875042521779652762, 7.47173950046536976078516747258, 7.74555136436794770677100913647, 7.991258178551428809923930705927, 8.236570732183097282250342914697, 8.370610222134724640929864522863, 8.640224288404582557307461277710
