| L(s) = 1 | − 108·3-s − 1.25e3·5-s + 908·7-s − 4.12e3·9-s − 2.51e4·11-s + 1.46e5·13-s + 1.35e5·15-s + 1.69e5·17-s + 2.54e4·19-s − 9.80e4·21-s − 1.78e6·23-s + 1.17e6·25-s + 2.43e4·27-s + 7.32e6·29-s − 1.06e7·31-s + 2.71e6·33-s − 1.13e6·35-s − 5.75e6·37-s − 1.58e7·39-s − 7.79e6·41-s − 1.67e7·43-s + 5.15e6·45-s − 1.53e7·47-s − 7.56e7·49-s − 1.82e7·51-s − 5.85e7·53-s + 3.14e7·55-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 0.894·5-s + 0.142·7-s − 0.209·9-s − 0.517·11-s + 1.42·13-s + 0.688·15-s + 0.491·17-s + 0.0448·19-s − 0.110·21-s − 1.32·23-s + 3/5·25-s + 0.00879·27-s + 1.92·29-s − 2.07·31-s + 0.398·33-s − 0.127·35-s − 0.504·37-s − 1.09·39-s − 0.430·41-s − 0.748·43-s + 0.187·45-s − 0.460·47-s − 1.87·49-s − 0.378·51-s − 1.01·53-s + 0.462·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 4 p^{3} T + 1754 p^{2} T^{2} + 4 p^{12} T^{3} + p^{18} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 908 T + 10919358 p T^{2} - 908 p^{9} T^{3} + p^{18} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 25120 T + 4397886806 T^{2} + 25120 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 146948 T + 18650572638 T^{2} - 146948 p^{9} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 169268 T + 244287370694 T^{2} - 169268 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 25480 T + 371498689782 T^{2} - 25480 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 1782748 T + 3965893132658 T^{2} + 1782748 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 7323340 T + 29495257443614 T^{2} - 7323340 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 10677272 T + 73017326150862 T^{2} + 10677272 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5750460 T + 104099098360718 T^{2} + 5750460 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 7795764 T + 505248245475190 T^{2} + 7795764 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 16770524 T + 1074543668703834 T^{2} + 16770524 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 15393892 T + 1097719682118050 T^{2} + 15393892 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 58529292 T + 3439533688251982 T^{2} + 58529292 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 59618264 T + 11433756193805126 T^{2} + 59618264 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3084652 p T + 22324076261167422 T^{2} + 3084652 p^{10} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 105998252 T + 21995694557368746 T^{2} - 105998252 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 168665592 T + 68474091725761822 T^{2} - 168665592 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 390212412 T + 130694746296409238 T^{2} + 390212412 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 466946256 T + 276599246367485918 T^{2} + 466946256 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 329535164 T + 370572645121965674 T^{2} + 329535164 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 108334860 T - 479764388871867818 T^{2} + 108334860 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2043058628 T + 2509217520410601030 T^{2} - 2043058628 p^{9} T^{3} + p^{18} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.95764571861674912826754333103, −11.85592749226455355071804265005, −10.99116435908604009369657361244, −10.95769529699130036615814442694, −10.17404396224519152151585015490, −9.573682031828391743319541525536, −8.501387941771958036599842709686, −8.438354912740563972760512145551, −7.69027713674757453546009752613, −7.06497412550734060731152968633, −6.11154828455073723897869879396, −5.96450799659414777147835591559, −4.99426723584683477078153551061, −4.49357630251100041568949142021, −3.47194786033776950879814593948, −3.21949731320385847346022443007, −1.86024216509524479998156123201, −1.18736901032669014144868414064, 0, 0,
1.18736901032669014144868414064, 1.86024216509524479998156123201, 3.21949731320385847346022443007, 3.47194786033776950879814593948, 4.49357630251100041568949142021, 4.99426723584683477078153551061, 5.96450799659414777147835591559, 6.11154828455073723897869879396, 7.06497412550734060731152968633, 7.69027713674757453546009752613, 8.438354912740563972760512145551, 8.501387941771958036599842709686, 9.573682031828391743319541525536, 10.17404396224519152151585015490, 10.95769529699130036615814442694, 10.99116435908604009369657361244, 11.85592749226455355071804265005, 11.95764571861674912826754333103
