| L(s) = 1 | − 2-s − 4·3-s − 6·5-s + 4·6-s + 2·7-s + 13·9-s + 6·10-s − 11-s − 4·13-s − 2·14-s + 24·15-s + 2·17-s − 13·18-s − 5·19-s − 8·21-s + 22-s − 4·23-s + 25·25-s + 4·26-s − 30·27-s + 10·29-s − 24·30-s − 2·31-s + 32-s + 4·33-s − 2·34-s − 12·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.30·3-s − 2.68·5-s + 1.63·6-s + 0.755·7-s + 13/3·9-s + 1.89·10-s − 0.301·11-s − 1.10·13-s − 0.534·14-s + 6.19·15-s + 0.485·17-s − 3.06·18-s − 1.14·19-s − 1.74·21-s + 0.213·22-s − 0.834·23-s + 5·25-s + 0.784·26-s − 5.77·27-s + 1.85·29-s − 4.38·30-s − 0.359·31-s + 0.176·32-s + 0.696·33-s − 0.342·34-s − 2.02·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234256 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05903088436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05903088436\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_2^2:C_4$ | \( 1 + 4 T + p T^{2} - 10 T^{3} - 29 T^{4} - 10 p T^{5} + p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 + 6 T + 11 T^{2} + 6 T^{3} + T^{4} + 6 p T^{5} + 11 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 - 2 T - 3 T^{2} + 20 T^{3} - 19 T^{4} + 20 p T^{5} - 3 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 4 T + 3 T^{2} + 50 T^{3} + 341 T^{4} + 50 p T^{5} + 3 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 2 T - 13 T^{2} - 20 T^{3} + 341 T^{4} - 20 p T^{5} - 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 5 T + 21 T^{2} + 145 T^{3} + 956 T^{4} + 145 p T^{5} + 21 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_4\times C_2$ | \( 1 - 10 T + 31 T^{2} - 200 T^{3} + 1821 T^{4} - 200 p T^{5} + 31 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 + 2 T - 27 T^{2} - 116 T^{3} + 605 T^{4} - 116 p T^{5} - 27 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 18 T + 107 T^{2} + 210 T^{3} + T^{4} + 210 p T^{5} + 107 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 2 T - 17 T^{2} - 236 T^{3} + 525 T^{4} - 236 p T^{5} - 17 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 3 T - 13 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 8 T + 17 T^{2} + 380 T^{3} + 4721 T^{4} + 380 p T^{5} + 17 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 4 T + 43 T^{2} + 380 T^{3} + 4761 T^{4} + 380 p T^{5} + 43 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 5 T + T^{2} + 335 T^{3} - 1164 T^{4} + 335 p T^{5} + p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 8 T + 3 T^{2} - 436 T^{3} + 6905 T^{4} - 436 p T^{5} + 3 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 11 T + 133 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 8 T - 47 T^{2} + 434 T^{3} + 1365 T^{4} + 434 p T^{5} - 47 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 14 T + 63 T^{2} - 500 T^{3} - 10579 T^{4} - 500 p T^{5} + 63 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 + 30 T + 461 T^{2} + 5400 T^{3} + 52861 T^{4} + 5400 p T^{5} + 461 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 19 T + 103 T^{2} + 695 T^{3} + 10536 T^{4} + 695 p T^{5} + 103 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 5 T + 153 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 3 T + 47 T^{2} - 255 T^{3} + 3496 T^{4} - 255 p T^{5} + 47 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−14.21927936304491680625721920771, −12.94620107202840274194465400940, −12.78371803078076126247183040055, −12.65389133375843142327103425648, −12.36241295443237250859504660591, −11.84533436676177067387993792360, −11.73196195975361073076144697363, −11.44619607659213608835038775453, −11.19015273008704886636155042019, −10.43192667523290300547890412354, −10.40424140172056982202672856051, −10.26581297587051140598940030101, −9.732900602864643249518309140274, −8.768998469539068183496117711620, −8.578299937453888876233453362782, −8.119000129251193742156669325387, −7.75096726427605455178351016739, −7.05504734034573282193115387091, −7.02176996947940726846070608250, −6.74059272803428382070771594247, −5.74400588385165905372239236141, −5.09333879761163886424786304608, −4.60558888920413881633024178031, −4.44474181454379931595270600981, −3.65206342848791883968556909270,
3.65206342848791883968556909270, 4.44474181454379931595270600981, 4.60558888920413881633024178031, 5.09333879761163886424786304608, 5.74400588385165905372239236141, 6.74059272803428382070771594247, 7.02176996947940726846070608250, 7.05504734034573282193115387091, 7.75096726427605455178351016739, 8.119000129251193742156669325387, 8.578299937453888876233453362782, 8.768998469539068183496117711620, 9.732900602864643249518309140274, 10.26581297587051140598940030101, 10.40424140172056982202672856051, 10.43192667523290300547890412354, 11.19015273008704886636155042019, 11.44619607659213608835038775453, 11.73196195975361073076144697363, 11.84533436676177067387993792360, 12.36241295443237250859504660591, 12.65389133375843142327103425648, 12.78371803078076126247183040055, 12.94620107202840274194465400940, 14.21927936304491680625721920771
