| L(s) = 1 | + 20·4-s + 90·5-s + 90·9-s + 504·11-s − 624·16-s + 440·19-s + 1.80e3·20-s + 4.97e3·25-s − 1.38e4·29-s − 1.35e4·31-s + 1.80e3·36-s + 396·41-s + 1.00e4·44-s + 8.10e3·45-s + 4.53e4·55-s + 4.93e4·59-s + 1.13e4·61-s − 3.29e4·64-s + 1.06e5·71-s + 8.80e3·76-s + 1.03e5·79-s − 5.61e4·80-s − 5.09e4·81-s + 1.99e4·89-s + 3.96e4·95-s + 4.53e4·99-s + 9.95e4·100-s + ⋯ |
| L(s) = 1 | + 5/8·4-s + 1.60·5-s + 0.370·9-s + 1.25·11-s − 0.609·16-s + 0.279·19-s + 1.00·20-s + 1.59·25-s − 3.06·29-s − 2.52·31-s + 0.231·36-s + 0.0367·41-s + 0.784·44-s + 0.596·45-s + 2.02·55-s + 1.84·59-s + 0.392·61-s − 1.00·64-s + 2.51·71-s + 0.174·76-s + 1.87·79-s − 0.981·80-s − 0.862·81-s + 0.267·89-s + 0.450·95-s + 0.465·99-s + 0.994·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.038612230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.038612230\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 - 18 p T + p^{5} T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 5 p^{2} T^{2} + p^{10} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 8 p T + p^{5} T^{2} )( 1 + 8 p T + p^{5} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 252 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 728330 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2363810 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 220 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6946370 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6930 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6752 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 56462470 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 198 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 293842250 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 347593490 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 802472090 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 24660 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5698 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 795787610 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 53352 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 883886830 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 51920 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4053674810 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9990 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 6923133890 T^{2} + p^{10} T^{4} \) |
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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
−11.61492400488104118989051106695, −10.89878479553880040152770734111, −10.66938773720813264628867435279, −9.872722327872960041377385898405, −9.501933812801325290737587307117, −9.084618099219475019334436097832, −8.990206537984244331858018170330, −7.907116925196730367800649140759, −7.35837584168313849192961712444, −6.79333607068090278460773611092, −6.62287456979023731019525533561, −5.68701279270296160301571207986, −5.62008010739812713787098889479, −4.90495009888309641192672677415, −3.76229697439406926303936169665, −3.69624693012619572428665713867, −2.45071780976177305158568266000, −1.77659665108182358239221945194, −1.76357753476231496978551879186, −0.60659669344056432807751757857,
0.60659669344056432807751757857, 1.76357753476231496978551879186, 1.77659665108182358239221945194, 2.45071780976177305158568266000, 3.69624693012619572428665713867, 3.76229697439406926303936169665, 4.90495009888309641192672677415, 5.62008010739812713787098889479, 5.68701279270296160301571207986, 6.62287456979023731019525533561, 6.79333607068090278460773611092, 7.35837584168313849192961712444, 7.907116925196730367800649140759, 8.990206537984244331858018170330, 9.084618099219475019334436097832, 9.501933812801325290737587307117, 9.872722327872960041377385898405, 10.66938773720813264628867435279, 10.89878479553880040152770734111, 11.61492400488104118989051106695
