| L(s) = 1 | − 2.04e3·4-s + 3.95e3·5-s − 1.18e5·9-s − 4.07e5·11-s + 3.14e6·16-s − 4.95e6·19-s − 8.09e6·20-s + 5.84e6·25-s + 2.41e8·36-s + 8.33e8·44-s − 4.66e8·45-s + 4.92e8·49-s − 1.60e9·55-s + 3.21e9·61-s − 4.29e9·64-s + 1.01e10·76-s + 1.24e10·80-s + 1.04e10·81-s − 1.95e10·95-s + 4.80e10·99-s − 1.19e10·100-s + 1.99e10·101-s + 7.23e10·121-s − 1.54e10·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2·4-s + 1.26·5-s − 2·9-s − 2.52·11-s + 3·16-s − 2·19-s − 2.52·20-s + 0.598·25-s + 4·36-s + 5.05·44-s − 2.52·45-s + 1.74·49-s − 3.19·55-s + 3.80·61-s − 4·64-s + 4·76-s + 3.79·80-s + 3·81-s − 2.52·95-s + 5.05·99-s − 1.19·100-s + 1.90·101-s + 2.79·121-s − 0.507·125-s − 6·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+5)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.4590216209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4590216209\) |
| \(L(6)\) |
|
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 | $C_2$ | \( 1 - 3951 T + p^{10} T^{2} \) |
| 19 | $C_1$ | \( ( 1 + p^{5} T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 32525 T + p^{10} T^{2} )( 1 + 32525 T + p^{10} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 203523 T + p^{10} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2146575 T + p^{10} T^{2} )( 1 + 2146575 T + p^{10} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 5138850 T + p^{10} T^{2} )( 1 + 5138850 T + p^{10} T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 212457925 T + p^{10} T^{2} )( 1 + 212457925 T + p^{10} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 456682125 T + p^{10} T^{2} )( 1 + 456682125 T + p^{10} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 1606836977 T + p^{10} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 3143217625 T + p^{10} T^{2} )( 1 + 3143217625 T + p^{10} T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 2150739450 T + p^{10} T^{2} )( 1 + 2150739450 T + p^{10} T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
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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
−12.60068889380601118170171174584, −11.77719259641969960674549637429, −10.76794745697115588505466805548, −10.63396366213544073879169660943, −9.991202521767680121983389418604, −9.600166445566417064624825380321, −8.676299838494493369399258466472, −8.595756068064213234994245972359, −8.231373408226320965486263192293, −7.45176617241802238810016800197, −6.17217720397381422258065391455, −5.80047099553414431823568676264, −5.15730103531334606385314228817, −5.14378654070056368628474526800, −4.13208760491157723585707550606, −3.31919316396098611609328388716, −2.42135042513146881104795493235, −2.27361095777284880183677926131, −0.75020506806192037575935143114, −0.24820434471412867367521316336,
0.24820434471412867367521316336, 0.75020506806192037575935143114, 2.27361095777284880183677926131, 2.42135042513146881104795493235, 3.31919316396098611609328388716, 4.13208760491157723585707550606, 5.14378654070056368628474526800, 5.15730103531334606385314228817, 5.80047099553414431823568676264, 6.17217720397381422258065391455, 7.45176617241802238810016800197, 8.231373408226320965486263192293, 8.595756068064213234994245972359, 8.676299838494493369399258466472, 9.600166445566417064624825380321, 9.991202521767680121983389418604, 10.63396366213544073879169660943, 10.76794745697115588505466805548, 11.77719259641969960674549637429, 12.60068889380601118170171174584
