| L(s) = 1 | − 1.02e3·4-s − 5.90e4·9-s − 1.07e6·11-s + 1.04e6·16-s + 1.31e7·19-s − 2.57e6·29-s − 1.00e8·31-s + 6.04e7·36-s − 1.35e9·41-s + 1.10e9·44-s + 3.09e9·49-s + 4.39e9·59-s + 1.98e10·61-s − 1.07e9·64-s + 3.50e10·71-s − 1.35e10·76-s + 2.73e9·79-s + 3.48e9·81-s + 1.29e11·89-s + 6.36e10·99-s − 2.99e11·101-s − 9.03e10·109-s + 2.63e9·116-s + 2.99e11·121-s + 1.03e11·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 2.01·11-s + 1/4·16-s + 1.22·19-s − 0.0233·29-s − 0.631·31-s + 1/6·36-s − 1.82·41-s + 1.00·44-s + 1.56·49-s + 0.799·59-s + 3.00·61-s − 1/8·64-s + 2.30·71-s − 0.611·76-s + 0.100·79-s + 1/9·81-s + 2.46·89-s + 0.672·99-s − 2.83·101-s − 0.562·109-s + 0.0116·116-s + 1.05·121-s + 0.315·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.268151028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.268151028\) |
| \(L(\frac{13}{2})\) |
|
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 | $C_2$ | \( 1 + p^{10} T^{2} \) |
| 3 | $C_2$ | \( 1 + p^{10} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 3093348382 T^{2} + p^{22} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 538680 T + p^{11} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3583665120838 T^{2} + p^{22} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 27636633878978 T^{2} + p^{22} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6599596 T + p^{11} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 680045448587854 T^{2} + p^{22} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 1288146 T + p^{11} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 50347432 T + p^{11} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 66883046829474410 T^{2} + p^{22} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 678700566 T + p^{11} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 1732577181193624150 T^{2} + p^{22} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 3465780428453834206 T^{2} + p^{22} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2315353969649815850 T^{2} + p^{22} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 2196450120 T + p^{11} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 9925999550 T + p^{11} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - \)\(24\!\cdots\!22\)\( T^{2} + p^{22} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 17538228960 T + p^{11} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - \)\(24\!\cdots\!58\)\( T^{2} + p^{22} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1369906648 T + p^{11} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + \)\(12\!\cdots\!22\)\( T^{2} + p^{22} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 64990633758 T + p^{11} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - \)\(40\!\cdots\!10\)\( T^{2} + p^{22} T^{4} \) |
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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
−11.07948428121850176415127586668, −10.66018987274215328968772415730, −10.12223288021957150570746673845, −9.799179090676958254304544023121, −9.142671822268279368351747233761, −8.602897774166331224795237643864, −7.979540198669331054307427891918, −7.83422185497971845910783694203, −7.02773042998104800233377820398, −6.59712871573258532387561546449, −5.49211425092502640581345108785, −5.32665368479449474602373178545, −5.10866799603819824266865121182, −4.05769254528782319258157417925, −3.57444586638891004604258954507, −2.87080147225612097379989098763, −2.39221217877523683379028353064, −1.71572983772996465764553102892, −0.65283199407993306595210988940, −0.49838267146243448985034950345,
0.49838267146243448985034950345, 0.65283199407993306595210988940, 1.71572983772996465764553102892, 2.39221217877523683379028353064, 2.87080147225612097379989098763, 3.57444586638891004604258954507, 4.05769254528782319258157417925, 5.10866799603819824266865121182, 5.32665368479449474602373178545, 5.49211425092502640581345108785, 6.59712871573258532387561546449, 7.02773042998104800233377820398, 7.83422185497971845910783694203, 7.979540198669331054307427891918, 8.602897774166331224795237643864, 9.142671822268279368351747233761, 9.799179090676958254304544023121, 10.12223288021957150570746673845, 10.66018987274215328968772415730, 11.07948428121850176415127586668
