| L(s) = 1 | + 2·5-s + 4·7-s − 3·9-s + 4·11-s + 4·13-s + 16·17-s + 8·19-s + 4·23-s + 25-s − 10·29-s − 4·31-s + 8·35-s − 6·41-s − 16·43-s − 6·45-s + 4·47-s + 15·49-s + 24·53-s + 8·55-s + 8·59-s + 10·61-s − 12·63-s + 8·65-s + 16·67-s − 24·71-s + 16·77-s + 24·79-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.51·7-s − 9-s + 1.20·11-s + 1.10·13-s + 3.88·17-s + 1.83·19-s + 0.834·23-s + 1/5·25-s − 1.85·29-s − 0.718·31-s + 1.35·35-s − 0.937·41-s − 2.43·43-s − 0.894·45-s + 0.583·47-s + 15/7·49-s + 3.29·53-s + 1.07·55-s + 1.04·59-s + 1.28·61-s − 1.51·63-s + 0.992·65-s + 1.95·67-s − 2.84·71-s + 1.82·77-s + 2.70·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.073781274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.073781274\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 4 T + T^{2} - 4 T^{3} + 64 T^{4} - 4 p T^{5} + p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 T + 2 T^{2} + 32 T^{3} - 101 T^{4} + 32 p T^{5} + 2 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T - 2 T^{2} + 32 T^{3} - 53 T^{4} + 32 p T^{5} - 2 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 - 4 T - 31 T^{2} - 4 T^{3} + 1312 T^{4} - 4 p T^{5} - 31 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p T^{2} )^{2}( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) |
| 31 | $C_2^2$ | \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 6 T - 7 T^{2} - 234 T^{3} - 1308 T^{4} - 234 p T^{5} - 7 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 16 T + 118 T^{2} + 832 T^{3} + 6187 T^{4} + 832 p T^{5} + 118 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 4 T - 79 T^{2} - 4 T^{3} + 6064 T^{4} - 4 p T^{5} - 79 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 - 8 T + 38 T^{2} + 736 T^{3} - 6581 T^{4} + 736 p T^{5} + 38 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 10 T - 35 T^{2} - 130 T^{3} + 8404 T^{4} - 130 p T^{5} - 35 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 16 T + 61 T^{2} - 976 T^{3} + 16384 T^{4} - 976 p T^{5} + 61 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 24 T + 286 T^{2} - 3168 T^{3} + 32355 T^{4} - 3168 p T^{5} + 286 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 16 T + 53 T^{2} + 592 T^{3} + 13072 T^{4} + 592 p T^{5} + 53 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 6 T + 139 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 4 T - 134 T^{2} + 176 T^{3} + 11539 T^{4} + 176 p T^{5} - 134 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.59991870314145153478055969415, −7.17490505221813749994428940928, −7.05748822572469880368412497714, −6.97179043446640638886547515686, −6.59447129674172733634572766983, −6.13113912867951698656711326412, −5.85377577678537829012567189287, −5.80359888194190095953867301198, −5.66276724249526166519175194963, −5.25155443306023009544516508255, −5.11656528281010800229079060664, −5.09109234301985860271388624928, −4.98745926398486597457892330879, −3.98767178447564564844249093783, −3.89745579472869260738067587030, −3.72267651682755436128150591237, −3.65573702992317032300555882914, −3.26373479817536228400367575992, −2.88065150988372629488412583131, −2.50941293676367291269323027473, −2.26106290758426829005800491590, −1.52129200964677987526090258954, −1.35318532116602086919910884241, −1.30804776938713247561657938505, −0.805421824636803346274993369744,
0.805421824636803346274993369744, 1.30804776938713247561657938505, 1.35318532116602086919910884241, 1.52129200964677987526090258954, 2.26106290758426829005800491590, 2.50941293676367291269323027473, 2.88065150988372629488412583131, 3.26373479817536228400367575992, 3.65573702992317032300555882914, 3.72267651682755436128150591237, 3.89745579472869260738067587030, 3.98767178447564564844249093783, 4.98745926398486597457892330879, 5.09109234301985860271388624928, 5.11656528281010800229079060664, 5.25155443306023009544516508255, 5.66276724249526166519175194963, 5.80359888194190095953867301198, 5.85377577678537829012567189287, 6.13113912867951698656711326412, 6.59447129674172733634572766983, 6.97179043446640638886547515686, 7.05748822572469880368412497714, 7.17490505221813749994428940928, 7.59991870314145153478055969415
