| L(s) = 1 | + 9.86e4·13-s + 9.69e5·25-s + 4.20e6·37-s + 2.10e6·49-s − 6.59e7·61-s + 4.02e7·73-s + 3.65e8·97-s + 3.90e8·109-s + 8.07e8·121-s + ⋯ |
| L(s) = 1 | + 3.45·13-s + 2.48·25-s + 2.24·37-s + 0.364·49-s − 4.76·61-s + 1.41·73-s + 4.12·97-s + 2.76·109-s + 3.76·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(18.89737361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.89737361\) |
| \(L(5)\) |
|
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 | $C_2^2$ | \( ( 1 - 19392 p^{2} T^{2} + p^{16} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 150034 p T^{2} + p^{16} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 403558082 T^{2} + p^{16} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 24672 T + p^{8} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 13946194560 T^{2} + p^{16} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 9612555842 T^{2} + p^{16} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 110101722242 T^{2} + p^{16} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 55195949760 T^{2} + p^{16} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 609424258558 T^{2} + p^{16} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 1050910 T + p^{8} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 14893421569920 T^{2} + p^{16} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 3086376619202 T^{2} + p^{16} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 24832499577598 T^{2} + p^{16} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 121619527525440 T^{2} + p^{16} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 285021141735362 T^{2} + p^{16} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 16478302 T + p^{8} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 710528440307522 T^{2} + p^{16} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 1248790731350402 T^{2} + p^{16} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10061952 T + p^{8} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 2130662749413122 T^{2} + p^{16} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 2942705721883202 T^{2} + p^{16} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 123253927680 p^{2} T^{2} + p^{16} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 91346112 T + p^{8} T^{2} )^{4} \) |
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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
−7.32599610656967135624868094104, −6.67406944577731863652484387911, −6.36965529467102877032964601544, −6.36221887714414418530106315785, −6.28780721437342101524723164199, −6.03644769713313496980195557231, −5.51855814203840290262351480373, −5.41938361203415450327414186686, −5.09208041288483230923675349570, −4.53899053312811135999817903189, −4.47561743095971639914030466812, −4.17393790999962443392348907879, −4.13469156502962570912903845567, −3.26294061568844917276810487177, −3.25914636174393717433521000600, −3.19946455458237734409420643305, −3.06724577561348489858668012889, −2.32489966756036473742291309469, −2.02973056434434946361707202335, −1.72076541277680034770838236198, −1.47518280663491952265496679411, −0.966010601130690199016454342559, −0.916541557588665006975471781592, −0.57684460303216284207187201888, −0.47965077205143933522432083935,
0.47965077205143933522432083935, 0.57684460303216284207187201888, 0.916541557588665006975471781592, 0.966010601130690199016454342559, 1.47518280663491952265496679411, 1.72076541277680034770838236198, 2.02973056434434946361707202335, 2.32489966756036473742291309469, 3.06724577561348489858668012889, 3.19946455458237734409420643305, 3.25914636174393717433521000600, 3.26294061568844917276810487177, 4.13469156502962570912903845567, 4.17393790999962443392348907879, 4.47561743095971639914030466812, 4.53899053312811135999817903189, 5.09208041288483230923675349570, 5.41938361203415450327414186686, 5.51855814203840290262351480373, 6.03644769713313496980195557231, 6.28780721437342101524723164199, 6.36221887714414418530106315785, 6.36965529467102877032964601544, 6.67406944577731863652484387911, 7.32599610656967135624868094104
