| L(s) = 1 | − 3·3-s − 5·5-s + 2·9-s + 2·13-s + 15·15-s − 2·17-s − 4·19-s + 6·23-s + 10·25-s + 6·27-s + 10·29-s + 31-s − 6·37-s − 6·39-s + 9·41-s + 3·43-s − 10·45-s − 4·47-s + 6·51-s + 9·53-s + 12·57-s − 6·59-s − 5·61-s − 10·65-s + 67-s − 18·69-s − 14·71-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 2.23·5-s + 2/3·9-s + 0.554·13-s + 3.87·15-s − 0.485·17-s − 0.917·19-s + 1.25·23-s + 2·25-s + 1.15·27-s + 1.85·29-s + 0.179·31-s − 0.986·37-s − 0.960·39-s + 1.40·41-s + 0.457·43-s − 1.49·45-s − 0.583·47-s + 0.840·51-s + 1.23·53-s + 1.58·57-s − 0.781·59-s − 0.640·61-s − 1.24·65-s + 0.122·67-s − 2.16·69-s − 1.66·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 7 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_4$ | \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - T - 39 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 9 T + 91 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 77 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9 T + 65 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 122 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 27 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T + 103 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 14 T + 186 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T + 135 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 138 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 22 T + 294 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 183 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 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
−8.254292534484650976710654540478, −8.249094780469195592028243824111, −7.57206372659170439310384138594, −7.35623950739432264766386568065, −6.77171094930393909700125203646, −6.67668351713048585785681933165, −6.15057464773386584460006655295, −5.90966369117195673871220615235, −5.32131417293714511684359432124, −4.98642304413560119702008758584, −4.54330346094629470107763660239, −4.32078110942679753195925306636, −3.82763275488386918786053268894, −3.48811960280432874215322560604, −2.71893385621336973296584060530, −2.60205011479295282947528503767, −1.32502125625042825062595647086, −0.921674295272300449676830037175, 0, 0,
0.921674295272300449676830037175, 1.32502125625042825062595647086, 2.60205011479295282947528503767, 2.71893385621336973296584060530, 3.48811960280432874215322560604, 3.82763275488386918786053268894, 4.32078110942679753195925306636, 4.54330346094629470107763660239, 4.98642304413560119702008758584, 5.32131417293714511684359432124, 5.90966369117195673871220615235, 6.15057464773386584460006655295, 6.67668351713048585785681933165, 6.77171094930393909700125203646, 7.35623950739432264766386568065, 7.57206372659170439310384138594, 8.249094780469195592028243824111, 8.254292534484650976710654540478
