| L(s) = 1 | − 2·2-s + 4·3-s + 4-s − 4·5-s − 8·6-s + 4·7-s + 2·8-s + 11·9-s + 8·10-s + 4·12-s + 2·13-s − 8·14-s − 16·15-s − 4·16-s + 18·17-s − 22·18-s + 10·19-s − 4·20-s + 16·21-s + 24·23-s + 8·24-s + 5·25-s − 4·26-s + 20·27-s + 4·28-s − 12·29-s + 32·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 2.30·3-s + 1/2·4-s − 1.78·5-s − 3.26·6-s + 1.51·7-s + 0.707·8-s + 11/3·9-s + 2.52·10-s + 1.15·12-s + 0.554·13-s − 2.13·14-s − 4.13·15-s − 16-s + 4.36·17-s − 5.18·18-s + 2.29·19-s − 0.894·20-s + 3.49·21-s + 5.00·23-s + 1.63·24-s + 25-s − 0.784·26-s + 3.84·27-s + 0.755·28-s − 2.22·29-s + 5.84·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.913794119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.913794119\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 T + 5 T^{2} + 4 T^{3} - 20 T^{4} + 4 p T^{5} + 5 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - 4 T + 8 T^{2} + 24 T^{3} - 97 T^{4} + 24 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 2 T - 11 T^{2} + 22 T^{3} + 4 T^{4} + 22 p T^{5} - 11 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 18 T + 168 T^{2} - 1080 T^{3} + 5147 T^{4} - 1080 p T^{5} + 168 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 10 T + 74 T^{2} - 384 T^{3} + 1871 T^{4} - 384 p T^{5} + 74 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 29 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 492 T^{3} + 3218 T^{4} + 492 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 10 T + 50 T^{2} - 60 T^{3} - 553 T^{4} - 60 p T^{5} + 50 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 12 T + 133 T^{2} + 1020 T^{3} + 7512 T^{4} + 1020 p T^{5} + 133 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} - 20 T^{3} - 2654 T^{4} - 20 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 16 T + 233 T^{2} - 2116 T^{3} + 18508 T^{4} - 2116 p T^{5} + 233 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 2 T + 50 T^{2} + 580 T^{3} - 209 T^{4} + 580 p T^{5} + 50 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 12 T + 36 T^{2} - 12 p T^{3} - 9433 T^{4} - 12 p^{2} T^{5} + 36 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 30 T + 306 T^{2} - 600 T^{3} - 7249 T^{4} - 600 p T^{5} + 306 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 8 T - 46 T^{2} - 256 T^{3} + 2515 T^{4} - 256 p T^{5} - 46 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 + 10 T + 50 T^{2} - 1080 T^{3} - 11641 T^{4} - 1080 p T^{5} + 50 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 38 T + 650 T^{2} + 6944 T^{3} + 62479 T^{4} + 6944 p T^{5} + 650 p^{2} T^{6} + 38 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 26 T + 194 T^{2} + 1204 T^{3} - 29153 T^{4} + 1204 p T^{5} + 194 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 356 T^{2} + 50310 T^{4} - 356 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.366248076188885033103603706685, −8.104065371039827063976707239735, −7.86834802018743895877839470914, −7.84669274081683740491010001061, −7.36364614950166980323497024033, −7.04888214283216669710351344405, −7.02043595614827467805826851103, −6.97707075817635144948493727973, −6.97669237460714715250315086268, −5.82956848961958141155756559045, −5.37526323787742226902464856525, −5.29861701337571979631364273943, −5.03973352116248190176634666391, −4.82440071725863633234951429884, −4.82032778253639872022704033537, −3.76554601353263863144179051282, −3.76086399644417182225675209113, −3.46771891406794368239209058427, −3.23191534241027447026541347727, −3.21479131348686224121636395301, −2.90372056659434525205651584198, −1.70140749900188379092676888726, −1.50732344114535191784529919917, −1.44099208712006002643621054645, −0.935111060567454286714856075917,
0.935111060567454286714856075917, 1.44099208712006002643621054645, 1.50732344114535191784529919917, 1.70140749900188379092676888726, 2.90372056659434525205651584198, 3.21479131348686224121636395301, 3.23191534241027447026541347727, 3.46771891406794368239209058427, 3.76086399644417182225675209113, 3.76554601353263863144179051282, 4.82032778253639872022704033537, 4.82440071725863633234951429884, 5.03973352116248190176634666391, 5.29861701337571979631364273943, 5.37526323787742226902464856525, 5.82956848961958141155756559045, 6.97669237460714715250315086268, 6.97707075817635144948493727973, 7.02043595614827467805826851103, 7.04888214283216669710351344405, 7.36364614950166980323497024033, 7.84669274081683740491010001061, 7.86834802018743895877839470914, 8.104065371039827063976707239735, 8.366248076188885033103603706685
