| L(s) = 1 | + 2·2-s + 2·4-s + 4·8-s + 8·16-s − 24·19-s − 2·25-s − 4·29-s + 24·31-s + 8·32-s − 24·37-s − 48·38-s − 2·49-s − 4·50-s − 8·53-s − 8·58-s + 48·62-s + 8·64-s − 48·74-s − 48·76-s + 48·83-s − 4·98-s − 4·100-s + 48·103-s − 16·106-s + 36·109-s − 8·113-s − 8·116-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.41·8-s + 2·16-s − 5.50·19-s − 2/5·25-s − 0.742·29-s + 4.31·31-s + 1.41·32-s − 3.94·37-s − 7.78·38-s − 2/7·49-s − 0.565·50-s − 1.09·53-s − 1.05·58-s + 6.09·62-s + 64-s − 5.57·74-s − 5.50·76-s + 5.26·83-s − 0.404·98-s − 2/5·100-s + 4.72·103-s − 1.55·106-s + 3.44·109-s − 0.752·113-s − 0.742·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.250390352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.250390352\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 11 | $D_4\times C_2$ | \( 1 - 30 T^{2} + 419 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $D_4\times C_2$ | \( 1 - 26 T^{2} + 315 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 2 T + 11 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 37 | $D_{4}$ | \( ( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 91 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 11190 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 228 T^{2} + 22310 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 7590 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 94 T^{2} + 3891 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 24 T + 298 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 31014 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 154 T^{2} + 20859 T^{4} - 154 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.70942207062398585066244545778, −6.46729655932502257039330089598, −6.39288908155171626429376229856, −6.34526710080354409511228400849, −6.12375857990017982652123914162, −5.98889347676214098991509499741, −5.55495079547206395839902428998, −5.21324941422193871365956030584, −4.91622927041094039839603390819, −4.80946036397823222361816509719, −4.66516084301488238001165598389, −4.48739208313487072455953898385, −4.38265057850529884408424905605, −3.84893615351133428777818958591, −3.71539924097507702492781206413, −3.62542349700719854761917951003, −3.30947402138291741896206331044, −2.96335085214019490004100720949, −2.44995537663784155786992805513, −2.31209634573333695917693431686, −1.99017824741381877206560464709, −1.83413695762571736137334436232, −1.64114303492814768542305209505, −0.76785247261405926814438718993, −0.40730130530226666974879295826,
0.40730130530226666974879295826, 0.76785247261405926814438718993, 1.64114303492814768542305209505, 1.83413695762571736137334436232, 1.99017824741381877206560464709, 2.31209634573333695917693431686, 2.44995537663784155786992805513, 2.96335085214019490004100720949, 3.30947402138291741896206331044, 3.62542349700719854761917951003, 3.71539924097507702492781206413, 3.84893615351133428777818958591, 4.38265057850529884408424905605, 4.48739208313487072455953898385, 4.66516084301488238001165598389, 4.80946036397823222361816509719, 4.91622927041094039839603390819, 5.21324941422193871365956030584, 5.55495079547206395839902428998, 5.98889347676214098991509499741, 6.12375857990017982652123914162, 6.34526710080354409511228400849, 6.39288908155171626429376229856, 6.46729655932502257039330089598, 6.70942207062398585066244545778
