| L(s) = 1 | − 16·4-s + 88·13-s + 192·16-s − 1.00e3·19-s + 2.37e3·25-s − 1.01e3·31-s − 928·37-s − 592·43-s + 686·49-s − 1.40e3·52-s − 1.25e4·61-s − 2.04e3·64-s + 1.68e4·67-s − 6.00e3·73-s + 1.60e4·76-s − 1.58e4·79-s + 5.27e4·97-s − 3.79e4·100-s − 3.09e4·103-s + 2.00e4·109-s − 7.40e3·121-s + 1.62e4·124-s + 127-s + 131-s + 137-s + 139-s + 1.48e4·148-s + ⋯ |
| L(s) = 1 | − 4-s + 0.520·13-s + 3/4·16-s − 2.77·19-s + 3.79·25-s − 1.05·31-s − 0.677·37-s − 0.320·43-s + 2/7·49-s − 0.520·52-s − 3.37·61-s − 1/2·64-s + 3.75·67-s − 1.12·73-s + 2.77·76-s − 2.53·79-s + 5.60·97-s − 3.79·100-s − 2.91·103-s + 1.68·109-s − 0.505·121-s + 1.05·124-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 0.677·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.442229376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.442229376\) |
| \(L(3)\) |
|
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 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 2372 T^{2} + 2185046 T^{4} - 2372 p^{8} T^{6} + p^{16} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 7408 T^{2} + 44053826 T^{4} + 7408 p^{8} T^{6} + p^{16} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 44 T + 54806 T^{2} - 44 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 155492 T^{2} + 19324224086 T^{4} - 155492 p^{8} T^{6} + p^{16} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 500 T + 212010 T^{2} + 500 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 507248 T^{2} + 194115433730 T^{4} - 507248 p^{8} T^{6} + p^{16} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2696528 T^{2} + 2815621714370 T^{4} - 2696528 p^{8} T^{6} + p^{16} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 508 T + 1273130 T^{2} + 508 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 464 T + 3357618 T^{2} + 464 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 10422916 T^{2} + 43126586993814 T^{4} - 10422916 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 296 T + 6229506 T^{2} + 296 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 3843028 T^{2} + 610858828518 T^{4} - 3843028 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 31511200 T^{2} + 372757754218434 T^{4} - 31511200 p^{8} T^{6} + p^{16} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 1937300 T^{2} + 227869699778342 T^{4} - 1937300 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 6280 T + 26046614 T^{2} + 6280 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 8436 T + 38430934 T^{2} - 8436 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 60749936 T^{2} + 2213591693878274 T^{4} - 60749936 p^{8} T^{6} + p^{16} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 3000 T + 33509782 T^{2} + 3000 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 100 p T + 36724934 T^{2} + 100 p^{5} T^{3} + p^{8} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 181804868 T^{2} + 12763622956082630 T^{4} - 181804868 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 28212772 T^{2} - 86397249754794 T^{4} - 28212772 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 26352 T + 344559046 T^{2} - 26352 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
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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
−9.038452742620995364699432600346, −8.735593249556021105776468429805, −8.723896995866102848796930713662, −8.310969915508543954246387957730, −8.244315655307792179747170530645, −7.63339734673133968949897202366, −7.40674362607871245497940827565, −6.98716921236462178382659467408, −6.65442399290120356520662936338, −6.59925525508125677202388712487, −6.00581513799457170710362174819, −5.97390822934961658408020116806, −5.29983059166381905957052996025, −5.12204784073315868159834129109, −4.68812769680752813254673395231, −4.48638421262843312383433688425, −4.18124056293598204300847765658, −3.75405222423016150107439024480, −3.16023288288749205138813422639, −3.08645147843145943931366383509, −2.44381834137479984557329885580, −1.81000757959368747908066024688, −1.51133771911047644930286786238, −0.60048577969609700973587408455, −0.49282594143390157734750070930,
0.49282594143390157734750070930, 0.60048577969609700973587408455, 1.51133771911047644930286786238, 1.81000757959368747908066024688, 2.44381834137479984557329885580, 3.08645147843145943931366383509, 3.16023288288749205138813422639, 3.75405222423016150107439024480, 4.18124056293598204300847765658, 4.48638421262843312383433688425, 4.68812769680752813254673395231, 5.12204784073315868159834129109, 5.29983059166381905957052996025, 5.97390822934961658408020116806, 6.00581513799457170710362174819, 6.59925525508125677202388712487, 6.65442399290120356520662936338, 6.98716921236462178382659467408, 7.40674362607871245497940827565, 7.63339734673133968949897202366, 8.244315655307792179747170530645, 8.310969915508543954246387957730, 8.723896995866102848796930713662, 8.735593249556021105776468429805, 9.038452742620995364699432600346
