| L(s) = 1 | − 2-s − 2·3-s − 2·4-s + 2·6-s + 7-s + 3·8-s + 3·9-s + 4·12-s + 3·13-s − 14-s + 16-s − 4·17-s − 3·18-s − 4·19-s − 2·21-s + 23-s − 6·24-s − 3·26-s − 4·27-s − 2·28-s + 4·29-s − 9·31-s − 2·32-s + 4·34-s − 6·36-s + 4·38-s − 6·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 4-s + 0.816·6-s + 0.377·7-s + 1.06·8-s + 9-s + 1.15·12-s + 0.832·13-s − 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.707·18-s − 0.917·19-s − 0.436·21-s + 0.208·23-s − 1.22·24-s − 0.588·26-s − 0.769·27-s − 0.377·28-s + 0.742·29-s − 1.61·31-s − 0.353·32-s + 0.685·34-s − 36-s + 0.648·38-s − 0.960·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 27 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - T + 35 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9 T + 71 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + T + 51 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 74 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 11 T + 105 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 77 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 126 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 153 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 165 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 122 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 5 T + 161 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 13 T + 159 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.52772105241633030692858886589, −7.22374090748217541658772592457, −6.85821353079416627717073130072, −6.61968128258821659625841597395, −6.09746488176982449427590445114, −5.88658562320704846461386067338, −5.53113400640190389431023886619, −5.03664217264099615366628609480, −4.89918949307519512403945891634, −4.36427730371600209474471360991, −4.17734036854244007851074448140, −3.88893099747149913243681216787, −3.38390233282866975920355723009, −2.80574578131278843305850354054, −2.12899623284332945453514524365, −1.88749901354293104812017015280, −1.13177001456849173018726428468, −0.941104436023682331557825960416, 0, 0,
0.941104436023682331557825960416, 1.13177001456849173018726428468, 1.88749901354293104812017015280, 2.12899623284332945453514524365, 2.80574578131278843305850354054, 3.38390233282866975920355723009, 3.88893099747149913243681216787, 4.17734036854244007851074448140, 4.36427730371600209474471360991, 4.89918949307519512403945891634, 5.03664217264099615366628609480, 5.53113400640190389431023886619, 5.88658562320704846461386067338, 6.09746488176982449427590445114, 6.61968128258821659625841597395, 6.85821353079416627717073130072, 7.22374090748217541658772592457, 7.52772105241633030692858886589
