| L(s) = 1 | + 96·7-s + 256·13-s + 1.15e3·19-s + 384·25-s + 2.78e3·31-s + 3.25e3·37-s + 8.64e3·43-s − 260·49-s − 312·61-s + 2.36e4·67-s − 1.61e4·73-s + 1.33e4·79-s + 2.45e4·91-s − 1.12e4·97-s − 1.16e4·103-s + 4.12e4·109-s − 1.23e4·121-s + ⋯ |
| L(s) = 1 | + 1.95·7-s + 1.51·13-s + 3.19·19-s + 0.614·25-s + 2.89·31-s + 2.37·37-s + 4.67·43-s − 0.108·49-s − 0.0838·61-s + 5.26·67-s − 3.02·73-s + 2.13·79-s + 2.96·91-s − 1.19·97-s − 1.09·103-s + 3.46·109-s − 0.843·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(17.81212718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.81212718\) |
| \(L(3)\) |
|
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 | $D_4\times C_2$ | \( 1 - 384 T^{2} + 237506 T^{4} - 384 p^{8} T^{6} + p^{16} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 48 T + 3586 T^{2} - 48 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 + 12348 T^{2} + 53959238 T^{4} + 12348 p^{8} T^{6} + p^{16} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 128 T + 45090 T^{2} - 128 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 265216 T^{2} + 31255382274 T^{4} - 265216 p^{8} T^{6} + p^{16} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 576 T + 314914 T^{2} - 576 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 776068 T^{2} + 289207442310 T^{4} - 776068 p^{8} T^{6} + p^{16} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2513280 T^{2} + 2575701102914 T^{4} - 2513280 p^{8} T^{6} + p^{16} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 1392 T + 1684546 T^{2} - 1392 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 44 p T + 3378726 T^{2} - 44 p^{5} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 10086912 T^{2} + 41102518325378 T^{4} - 10086912 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 4320 T + 10635874 T^{2} - 4320 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 11276292 T^{2} + 74962747331846 T^{4} - 11276292 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 25153920 T^{2} + 282144339893954 T^{4} - 25153920 p^{8} T^{6} + p^{16} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 16703940 T^{2} + 136346193182534 T^{4} - 16703940 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 156 T + 1892966 T^{2} + 156 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 11808 T + 74808226 T^{2} - 11808 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 74615940 T^{2} + 2507552925381254 T^{4} - 74615940 p^{8} T^{6} + p^{16} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 8064 T + 72795458 T^{2} + 8064 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 6672 T + 57330370 T^{2} - 6672 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 124074820 T^{2} + 7474823548026054 T^{4} - 124074820 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 118087680 T^{2} + 9299246444069762 T^{4} - 118087680 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 5632 T + 144668418 T^{2} + 5632 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
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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
−8.176060877169326385122990228922, −7.47666329197901989068533980198, −7.47528444898847306694446160379, −7.41022251165943010277904887297, −7.04815777040251040532416959985, −6.31518734873348132238768927266, −6.26019121523485301982399491907, −6.23063420008242089623771285401, −5.81683078939458778798964396183, −5.41875715329060683162186063084, −5.07824408624322082857181361162, −5.02272990493537731045781343619, −4.74327280351858575095232685348, −4.26760214271743083884639957943, −4.14561513524387269886734456895, −3.66498586312915585596217737302, −3.52275376743823434890563315796, −2.84165054426321504510424238169, −2.57471903497981976275715067270, −2.57081343448804584860151904746, −1.88572650030729782005226221959, −1.13837884837602899605472458966, −1.06841860949465032833585669376, −1.06103022728592977038858211282, −0.63540416840558061704136981457,
0.63540416840558061704136981457, 1.06103022728592977038858211282, 1.06841860949465032833585669376, 1.13837884837602899605472458966, 1.88572650030729782005226221959, 2.57081343448804584860151904746, 2.57471903497981976275715067270, 2.84165054426321504510424238169, 3.52275376743823434890563315796, 3.66498586312915585596217737302, 4.14561513524387269886734456895, 4.26760214271743083884639957943, 4.74327280351858575095232685348, 5.02272990493537731045781343619, 5.07824408624322082857181361162, 5.41875715329060683162186063084, 5.81683078939458778798964396183, 6.23063420008242089623771285401, 6.26019121523485301982399491907, 6.31518734873348132238768927266, 7.04815777040251040532416959985, 7.41022251165943010277904887297, 7.47528444898847306694446160379, 7.47666329197901989068533980198, 8.176060877169326385122990228922
