| L(s) = 1 | − 32·5-s + 296·13-s − 576·17-s − 420·25-s + 3.48e3·29-s − 1.32e3·37-s − 6.84e3·41-s + 3.04e3·49-s + 1.10e4·53-s − 424·61-s − 9.47e3·65-s + 3.00e3·73-s + 1.84e4·85-s + 384·89-s + 1.11e4·97-s − 2.91e3·101-s − 1.86e4·109-s − 2.17e3·113-s + 2.73e4·121-s + 2.06e4·125-s + 127-s + 131-s + 137-s + 139-s − 1.11e5·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.27·5-s + 1.75·13-s − 1.99·17-s − 0.671·25-s + 4.14·29-s − 0.964·37-s − 4.07·41-s + 1.26·49-s + 3.93·53-s − 0.113·61-s − 2.24·65-s + 0.562·73-s + 2.55·85-s + 0.0484·89-s + 1.18·97-s − 0.285·101-s − 1.56·109-s − 0.170·113-s + 1.86·121-s + 1.32·125-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s − 5.30·145-s + 4.50e−5·149-s + 4.38e−5·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.024749570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.024749570\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 + 16 T + 594 T^{2} + 16 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 3044 T^{2} + 9238086 T^{4} - 3044 p^{8} T^{6} + p^{16} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 27332 T^{2} + 426733638 T^{4} - 27332 p^{8} T^{6} + p^{16} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 148 T + 16518 T^{2} - 148 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 288 T + 184898 T^{2} + 288 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 497092 T^{2} + 265140438 p^{2} T^{4} - 497092 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 257156 T^{2} + 145975182726 T^{4} - 257156 p^{8} T^{6} + p^{16} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 1744 T + 2053266 T^{2} - 1744 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2943332 T^{2} + 3834092018118 T^{4} - 2943332 p^{8} T^{6} + p^{16} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 660 T + 2705222 T^{2} + 660 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 3424 T + 8095746 T^{2} + 3424 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6217796 T^{2} + 20023638876486 T^{4} - 6217796 p^{8} T^{6} + p^{16} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 3265276 T^{2} + 3102160162566 T^{4} + 3265276 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 5520 T + 19769042 T^{2} - 5520 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 4500932 T^{2} - 162959688247482 T^{4} - 4500932 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 212 T + 27288198 T^{2} + 212 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 15078212 T^{2} + 51214632004038 T^{4} - 15078212 p^{8} T^{6} + p^{16} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 24936836 T^{2} + 605069020552326 T^{4} - 24936836 p^{8} T^{6} + p^{16} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 1500 T + 57174662 T^{2} - 1500 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 45515876 T^{2} + 513196620838086 T^{4} - 45515876 p^{8} T^{6} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 134137412 T^{2} + 8408135231992518 T^{4} - 134137412 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 192 T + 30096578 T^{2} - 192 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 5564 T + 18910086 T^{2} - 5564 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.10364311526159162232956205315, −6.83661319604337201183771902517, −6.68152996252465701347829823437, −6.53367761667171016604929236170, −6.18306088866450410956309275031, −5.93987292310970383867578120861, −5.73196641218621133618866913293, −5.37245994531016912596533392964, −4.96490656240799548240047867849, −4.82073240164101012993042132706, −4.66757323540461697995246123801, −4.34900989722254306933718755603, −3.97943617581708250908151266122, −3.68452006901034150366874401981, −3.58626945256321745081695499714, −3.57024480761545314070358311486, −2.79956029217254809298925963391, −2.71503727398566380931777681940, −2.35567827203594710917487029798, −2.02873684657102339299461767295, −1.58140817667367819962053863891, −1.23672354537851830102786294923, −0.933474132295497378377006682350, −0.37787499620693977238192161524, −0.37739382310423123503559989241,
0.37739382310423123503559989241, 0.37787499620693977238192161524, 0.933474132295497378377006682350, 1.23672354537851830102786294923, 1.58140817667367819962053863891, 2.02873684657102339299461767295, 2.35567827203594710917487029798, 2.71503727398566380931777681940, 2.79956029217254809298925963391, 3.57024480761545314070358311486, 3.58626945256321745081695499714, 3.68452006901034150366874401981, 3.97943617581708250908151266122, 4.34900989722254306933718755603, 4.66757323540461697995246123801, 4.82073240164101012993042132706, 4.96490656240799548240047867849, 5.37245994531016912596533392964, 5.73196641218621133618866913293, 5.93987292310970383867578120861, 6.18306088866450410956309275031, 6.53367761667171016604929236170, 6.68152996252465701347829823437, 6.83661319604337201183771902517, 7.10364311526159162232956205315
