| L(s) = 1 | + 4·2-s + 10·4-s + 20·8-s + 8·11-s + 35·16-s + 32·22-s − 8·23-s + 16·29-s + 56·32-s + 8·37-s + 80·44-s − 32·46-s + 32·53-s + 64·58-s + 84·64-s − 20·67-s − 8·71-s + 32·74-s + 32·79-s − 7·81-s + 160·88-s − 80·92-s + 128·106-s − 4·107-s + 72·109-s + 16·113-s + 160·116-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s + 7.07·8-s + 2.41·11-s + 35/4·16-s + 6.82·22-s − 1.66·23-s + 2.97·29-s + 9.89·32-s + 1.31·37-s + 12.0·44-s − 4.71·46-s + 4.39·53-s + 8.40·58-s + 21/2·64-s − 2.44·67-s − 0.949·71-s + 3.71·74-s + 3.60·79-s − 7/9·81-s + 17.0·88-s − 8.34·92-s + 12.4·106-s − 0.386·107-s + 6.89·109-s + 1.50·113-s + 14.8·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(65.57330333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(65.57330333\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 + 7 T^{4} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 4 T^{2} + 166 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 8 T^{2} + 55 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 64 T^{2} + 1735 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 104 T^{2} + 5527 T^{4} + 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 140 T^{2} + 9142 T^{4} + 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 + 176 T^{2} + 14002 T^{4} + 176 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 196 T^{2} + 16870 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 10 T + 115 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 + 232 T^{2} + 23575 T^{4} + 232 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 16 T + 178 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 96 T^{2} + 13607 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 248 T^{2} + 30327 T^{4} + 248 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 176 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.36582846362665238614167838196, −6.05363332782385809563217991479, −5.89273455402983024099763734749, −5.68998758732969977973783499658, −5.67983985466182733671280561989, −5.26683608678059692154449679443, −4.89143535073419907777703123334, −4.86619709621644623339540985601, −4.62662959203316090623045354209, −4.39853394456205118854675243287, −4.18520596280213788272310170165, −4.08322559166624970160066470651, −3.93720044389625648617723619856, −3.65674352301530078913705398101, −3.34035057053279598058129844296, −3.20402372946152732104744306201, −3.07220704281233539984496649352, −2.50204672166952138292254899027, −2.43430764113280699477357491711, −2.25086598796570360234244804215, −1.93741036319594043216243300236, −1.58623123911420564553166759526, −1.23397469299771019266967023079, −0.843033582991977651636537017291, −0.75360706468112153573863342311,
0.75360706468112153573863342311, 0.843033582991977651636537017291, 1.23397469299771019266967023079, 1.58623123911420564553166759526, 1.93741036319594043216243300236, 2.25086598796570360234244804215, 2.43430764113280699477357491711, 2.50204672166952138292254899027, 3.07220704281233539984496649352, 3.20402372946152732104744306201, 3.34035057053279598058129844296, 3.65674352301530078913705398101, 3.93720044389625648617723619856, 4.08322559166624970160066470651, 4.18520596280213788272310170165, 4.39853394456205118854675243287, 4.62662959203316090623045354209, 4.86619709621644623339540985601, 4.89143535073419907777703123334, 5.26683608678059692154449679443, 5.67983985466182733671280561989, 5.68998758732969977973783499658, 5.89273455402983024099763734749, 6.05363332782385809563217991479, 6.36582846362665238614167838196
