| L(s) = 1 | + 32·4-s − 4·7-s + 5.90e3·13-s + 2.10e4·19-s − 2.45e3·25-s − 128·28-s − 4.57e4·31-s + 1.36e5·37-s + 1.28e4·43-s + 2.35e5·49-s + 1.88e5·52-s + 1.25e5·61-s − 3.27e4·64-s − 8.77e5·67-s − 2.92e6·73-s + 6.73e5·76-s − 6.81e5·79-s − 2.36e4·91-s + 5.62e5·97-s − 7.84e4·100-s + 1.73e6·103-s + 2.60e6·109-s − 3.54e6·121-s − 1.46e6·124-s + 127-s + 131-s − 8.41e4·133-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.0116·7-s + 2.68·13-s + 3.06·19-s − 0.156·25-s − 0.00583·28-s − 1.53·31-s + 2.68·37-s + 0.161·43-s + 2.00·49-s + 1.34·52-s + 0.551·61-s − 1/8·64-s − 2.91·67-s − 7.51·73-s + 1.53·76-s − 1.38·79-s − 0.0313·91-s + 0.615·97-s − 0.0783·100-s + 1.58·103-s + 2.01·109-s − 1.99·121-s − 0.768·124-s − 0.0357·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.1497325276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1497325276\) |
| \(L(4)\) |
|
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 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 + 98 p^{2} T^{2} - 381021 p^{4} T^{4} + 98 p^{14} T^{6} + p^{24} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T - 117645 T^{2} + 2 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 3541970 T^{2} + 9407123104179 T^{4} + 3541970 p^{12} T^{6} + p^{24} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 2950 T + 3875691 T^{2} - 2950 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 28202690 T^{2} + p^{12} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5258 T + p^{6} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 191004770 T^{2} + 14568197730732579 T^{4} + 191004770 p^{12} T^{6} + p^{24} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 1184779442 T^{2} + 1049887542980362323 T^{4} + 1184779442 p^{12} T^{6} + p^{24} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 22898 T - 363185277 T^{2} + 22898 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 34058 T + p^{6} T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 + 9219079010 T^{2} + 62427927492256394019 T^{4} + 9219079010 p^{12} T^{6} + p^{24} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 6406 T - 6280326213 T^{2} - 6406 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 10801249342 T^{2} + 475504239106854723 T^{4} - 10801249342 p^{12} T^{6} + p^{24} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 7253988050 T^{2} + p^{12} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 22449655150 T^{2} - \)\(12\!\cdots\!81\)\( T^{4} - 22449655150 p^{12} T^{6} + p^{24} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 62566 T - 47605870005 T^{2} - 62566 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 438698 T + 101997553035 T^{2} + 438698 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 251546372642 T^{2} + p^{12} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 730510 T + p^{6} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 340562 T - 127104979677 T^{2} + 340562 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 407613512306 T^{2} + \)\(59\!\cdots\!75\)\( T^{4} + 407613512306 p^{12} T^{6} + p^{24} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 844406214050 T^{2} + p^{12} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 281086 T - 753962665533 T^{2} - 281086 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
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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
−8.149969885635572941500068722082, −8.069889290010106088024150278538, −7.41322192915653128069404378863, −7.33656872592518306732021674596, −7.31711115646106264364133706410, −7.24426392605560812004831114944, −6.25437516295783102369798934076, −6.20991651488451576404913455519, −6.04084209406382272590562337587, −5.81829656950748077443397629521, −5.36914134116636138594873073585, −5.35409283033718394134411905096, −4.60314378439435611670798204338, −4.26712257480328775783731850027, −4.19905219592551561669395784487, −3.61673783899774643682870978292, −3.31538314437964801486988365689, −3.16827033764260781403080105285, −2.58120050985987963737727057501, −2.52920982396873063295103589816, −1.66987697180957268659005620923, −1.19628588494882062098064177755, −1.18935927989187356768736107792, −1.08655810898071995009183404923, −0.04227961373058927213914753941,
0.04227961373058927213914753941, 1.08655810898071995009183404923, 1.18935927989187356768736107792, 1.19628588494882062098064177755, 1.66987697180957268659005620923, 2.52920982396873063295103589816, 2.58120050985987963737727057501, 3.16827033764260781403080105285, 3.31538314437964801486988365689, 3.61673783899774643682870978292, 4.19905219592551561669395784487, 4.26712257480328775783731850027, 4.60314378439435611670798204338, 5.35409283033718394134411905096, 5.36914134116636138594873073585, 5.81829656950748077443397629521, 6.04084209406382272590562337587, 6.20991651488451576404913455519, 6.25437516295783102369798934076, 7.24426392605560812004831114944, 7.31711115646106264364133706410, 7.33656872592518306732021674596, 7.41322192915653128069404378863, 8.069889290010106088024150278538, 8.149969885635572941500068722082
