| L(s) = 1 | + 4·2-s + 7·4-s + 7·8-s − 11-s − 5·13-s + 7·16-s + 2·17-s − 3·19-s − 4·22-s + 8·23-s − 20·26-s + 7·29-s − 5·31-s + 14·32-s + 8·34-s − 5·37-s − 12·38-s − 41-s − 5·43-s − 7·44-s + 32·46-s + 3·47-s − 14·49-s − 35·52-s + 19·53-s + 28·58-s − 10·59-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 7/2·4-s + 2.47·8-s − 0.301·11-s − 1.38·13-s + 7/4·16-s + 0.485·17-s − 0.688·19-s − 0.852·22-s + 1.66·23-s − 3.92·26-s + 1.29·29-s − 0.898·31-s + 2.47·32-s + 1.37·34-s − 0.821·37-s − 1.94·38-s − 0.156·41-s − 0.762·43-s − 1.05·44-s + 4.71·46-s + 0.437·47-s − 2·49-s − 4.85·52-s + 2.60·53-s + 3.67·58-s − 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(18.07076922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.07076922\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $C_6$ | \( 1 - p^{2} T + 9 T^{2} - 15 T^{3} + 9 p T^{4} - p^{4} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 2 p T^{2} - p T^{3} + 2 p^{2} T^{4} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + T + 17 T^{2} + 35 T^{3} + 17 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 5 T + 45 T^{2} + 131 T^{3} + 45 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 2 T + 36 T^{2} - 81 T^{3} + 36 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 8 T + 88 T^{2} - 381 T^{3} + 88 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 7 T + 45 T^{2} - 315 T^{3} + 45 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 5 T + 57 T^{2} + 353 T^{3} + 57 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 5 T + 103 T^{2} + 371 T^{3} + 103 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + T + 9 T^{2} - 339 T^{3} + 9 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 5 T + 72 T^{2} + 137 T^{3} + 72 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 3 T + 137 T^{2} - 269 T^{3} + 137 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 19 T + 214 T^{2} - 1707 T^{3} + 214 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 10 T + 145 T^{2} + 852 T^{3} + 145 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 17 T + 249 T^{2} + 2033 T^{3} + 249 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - T + 59 T^{2} - 693 T^{3} + 59 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 19 T + 268 T^{2} - 2391 T^{3} + 268 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - T + 49 T^{2} + 317 T^{3} + 49 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 18 T + 324 T^{2} - 2941 T^{3} + 324 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 13 T + 247 T^{2} - 2019 T^{3} + 247 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 2 T + 168 T^{2} + 75 T^{3} + 168 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 5 T + 297 T^{2} + 971 T^{3} + 297 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.24455088512502562142293503430, −6.98590356714992552718481904634, −6.85193740281325428701340985297, −6.67183755498626942964999695973, −6.19282341387116049332115622354, −6.05479978319681183259460099830, −5.93872211806772234407242806608, −5.30117960859128182678004294660, −5.24990266022017163118141868418, −5.21411183120775026603201071002, −4.80423989237671462796517171126, −4.66529267025973110098059542926, −4.64937862756995684931921459804, −4.03892795128303882282703424575, −3.96346045918927458421004081203, −3.60840451625210235409717677486, −3.26111830200762177772592548768, −3.16934635859923543380146947119, −2.85056001120812631586795947774, −2.55913061931990186416603032040, −2.14916219650401031413050451141, −1.78753134693225341709791571982, −1.52816581099161486566518054274, −0.65607122455543860387982537649, −0.60033970535823511752694568237,
0.60033970535823511752694568237, 0.65607122455543860387982537649, 1.52816581099161486566518054274, 1.78753134693225341709791571982, 2.14916219650401031413050451141, 2.55913061931990186416603032040, 2.85056001120812631586795947774, 3.16934635859923543380146947119, 3.26111830200762177772592548768, 3.60840451625210235409717677486, 3.96346045918927458421004081203, 4.03892795128303882282703424575, 4.64937862756995684931921459804, 4.66529267025973110098059542926, 4.80423989237671462796517171126, 5.21411183120775026603201071002, 5.24990266022017163118141868418, 5.30117960859128182678004294660, 5.93872211806772234407242806608, 6.05479978319681183259460099830, 6.19282341387116049332115622354, 6.67183755498626942964999695973, 6.85193740281325428701340985297, 6.98590356714992552718481904634, 7.24455088512502562142293503430
