| L(s) = 1 | − 9·3-s + 142·7-s − 191·13-s − 46·19-s − 1.27e3·21-s − 625·25-s + 729·27-s + 3.11e3·31-s + 5.18e3·37-s + 1.71e3·39-s − 23·43-s + 1.03e4·49-s + 414·57-s + 5.23e3·61-s − 2.90e3·67-s + 1.24e3·73-s + 5.62e3·75-s + 1.23e4·79-s − 6.56e3·81-s − 2.71e4·91-s − 2.80e4·93-s + 1.88e4·97-s − 3.96e4·103-s − 2.20e4·109-s − 4.66e4·111-s + 2.92e4·121-s + 127-s + ⋯ |
| L(s) = 1 | − 3-s + 2.89·7-s − 1.13·13-s − 0.127·19-s − 2.89·21-s − 25-s + 27-s + 3.24·31-s + 3.78·37-s + 1.13·39-s − 0.0124·43-s + 4.29·49-s + 0.127·57-s + 1.40·61-s − 0.646·67-s + 0.234·73-s + 75-s + 1.98·79-s − 81-s − 3.27·91-s − 3.24·93-s + 1.99·97-s − 3.74·103-s − 1.85·109-s − 3.78·111-s + 2·121-s + 6.20e−5·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.831786875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.831786875\) |
| \(L(3)\) |
|
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 | $C_2$ | \( 1 + p^{2} T + p^{4} T^{2} \) |
| 19 | $C_2$ | \( 1 + 46 T + p^{4} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 71 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 146 T + p^{4} T^{2} )( 1 + 337 T + p^{4} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 1559 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2591 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 3191 T + p^{4} T^{2} )( 1 + 3214 T + p^{4} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 7199 T + p^{4} T^{2} )( 1 + 1966 T + p^{4} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5906 T + p^{4} T^{2} )( 1 + 8809 T + p^{4} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 9791 T + p^{4} T^{2} )( 1 + 8542 T + p^{4} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 7682 T + p^{4} T^{2} )( 1 - 4679 T + p^{4} T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 9743 T + p^{4} T^{2} )( 1 - 9071 T + p^{4} T^{2} ) \) |
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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
−11.62394706780008393715974952801, −11.43763460706441927905717507499, −11.00239079562871987721590959329, −10.61824708230175526729233150788, −9.767244955174765215884344642748, −9.696906605908838512420665779566, −8.565118411642556836104576481178, −8.279674393963340989374786830938, −7.68769824053323654250359223473, −7.68746289462272042035094896217, −6.55037477937524611438749477092, −6.16671743979012587566907643276, −5.39243124610598247153971990022, −5.02911826588572155246240210756, −4.37292297045494911369830414498, −4.36848207073036177364206658833, −2.66383074662197398236043423859, −2.25621252336581142610741909336, −1.20770429599736358604065221798, −0.70789519464333782668021878546,
0.70789519464333782668021878546, 1.20770429599736358604065221798, 2.25621252336581142610741909336, 2.66383074662197398236043423859, 4.36848207073036177364206658833, 4.37292297045494911369830414498, 5.02911826588572155246240210756, 5.39243124610598247153971990022, 6.16671743979012587566907643276, 6.55037477937524611438749477092, 7.68746289462272042035094896217, 7.68769824053323654250359223473, 8.279674393963340989374786830938, 8.565118411642556836104576481178, 9.696906605908838512420665779566, 9.767244955174765215884344642748, 10.61824708230175526729233150788, 11.00239079562871987721590959329, 11.43763460706441927905717507499, 11.62394706780008393715974952801
