| L(s) = 1 | − 5-s + 2·7-s − 5·9-s + 7·11-s − 2·13-s + 3·17-s − 4·19-s + 5·23-s − 9·25-s + 27-s − 29-s − 9·31-s − 2·35-s − 37-s − 12·41-s − 18·43-s + 5·45-s + 16·47-s − 6·49-s + 5·53-s − 7·55-s − 3·59-s − 12·61-s − 10·63-s + 2·65-s + 13·67-s − 12·71-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s − 5/3·9-s + 2.11·11-s − 0.554·13-s + 0.727·17-s − 0.917·19-s + 1.04·23-s − 9/5·25-s + 0.192·27-s − 0.185·29-s − 1.61·31-s − 0.338·35-s − 0.164·37-s − 1.87·41-s − 2.74·43-s + 0.745·45-s + 2.33·47-s − 6/7·49-s + 0.686·53-s − 0.943·55-s − 0.390·59-s − 1.53·61-s − 1.25·63-s + 0.248·65-s + 1.58·67-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 17^{3} \cdot 59^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 17^{3} \cdot 59^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.204408010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.204408010\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - T )^{3} \) |
| 59 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 5 T^{2} - T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + T + 2 p T^{2} + 12 T^{3} + 2 p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 2 T + 10 T^{2} - 32 T^{3} + 10 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 7 T + 34 T^{2} - 108 T^{3} + 34 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 2 T + 35 T^{2} + 45 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 12 T^{2} - 44 T^{3} + 12 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 5 T + 45 T^{2} - 234 T^{3} + 45 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + T + 55 T^{2} - 6 T^{3} + 55 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 9 T + 104 T^{2} + 529 T^{3} + 104 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + T + 72 T^{2} + 4 p T^{3} + 72 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 12 T + 155 T^{2} + 976 T^{3} + 155 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 18 T + 134 T^{2} + 752 T^{3} + 134 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 16 T + 176 T^{2} - 1256 T^{3} + 176 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 5 T + 86 T^{2} - 646 T^{3} + 86 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 12 T + 38 T^{2} - 248 T^{3} + 38 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 13 T + 236 T^{2} - 1701 T^{3} + 236 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 12 T + 218 T^{2} + 1490 T^{3} + 218 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 10 T + 30 T^{2} - 134 T^{3} + 30 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - T + 58 T^{2} - 52 T^{3} + 58 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 2 T + 99 T^{2} + 477 T^{3} + 99 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 14 T + 320 T^{2} + 2548 T^{3} + 320 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 14 T + 325 T^{2} - 2663 T^{3} + 325 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.07481652260056025125260715351, −6.58348168026625337533691815337, −6.32573830058510742935966660822, −6.28899425447825095830219816972, −6.05690815661168990758896176141, −5.82462457531701816777968904255, −5.52082896635110803033918822312, −5.13814639401748294049765645740, −5.01231943333976454244110561340, −4.93262255427718798769131566976, −4.64625115263311058651110471528, −4.08663157558775205004273874643, −4.07604033500828860875953860251, −3.58194924016842903184145787649, −3.58098428256792590818121875588, −3.48060833395690423363327331912, −2.97268687044539254714763968308, −2.66271722050677714733675767016, −2.47000841217195423636613700696, −1.85702502584451362014801075100, −1.68045353529630947344762400160, −1.67048122252244862284718893225, −1.19970569679073213795511279765, −0.41798401510573906133530133201, −0.39461782252199158545793799651,
0.39461782252199158545793799651, 0.41798401510573906133530133201, 1.19970569679073213795511279765, 1.67048122252244862284718893225, 1.68045353529630947344762400160, 1.85702502584451362014801075100, 2.47000841217195423636613700696, 2.66271722050677714733675767016, 2.97268687044539254714763968308, 3.48060833395690423363327331912, 3.58098428256792590818121875588, 3.58194924016842903184145787649, 4.07604033500828860875953860251, 4.08663157558775205004273874643, 4.64625115263311058651110471528, 4.93262255427718798769131566976, 5.01231943333976454244110561340, 5.13814639401748294049765645740, 5.52082896635110803033918822312, 5.82462457531701816777968904255, 6.05690815661168990758896176141, 6.28899425447825095830219816972, 6.32573830058510742935966660822, 6.58348168026625337533691815337, 7.07481652260056025125260715351
