| L(s) = 1 | + 2·3-s − 2·5-s − 4·7-s + 3·9-s + 4·11-s − 4·13-s − 4·15-s − 8·21-s + 3·25-s + 4·27-s − 4·29-s − 8·31-s + 8·33-s + 8·35-s + 4·37-s − 8·39-s + 4·41-s − 8·43-s − 6·45-s − 8·47-s − 2·49-s − 12·53-s − 8·55-s + 4·59-s + 8·61-s − 12·63-s + 8·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 1.51·7-s + 9-s + 1.20·11-s − 1.10·13-s − 1.03·15-s − 1.74·21-s + 3/5·25-s + 0.769·27-s − 0.742·29-s − 1.43·31-s + 1.39·33-s + 1.35·35-s + 0.657·37-s − 1.28·39-s + 0.624·41-s − 1.21·43-s − 0.894·45-s − 1.16·47-s − 2/7·49-s − 1.64·53-s − 1.07·55-s + 0.520·59-s + 1.02·61-s − 1.51·63-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 114 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 106 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 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
−8.162244655544744862092529867049, −8.105500556012422632035757697208, −7.48206244061132668655557186412, −7.26267370186674390213852162645, −6.87746432065015431590065997132, −6.72299191914191127814883950879, −6.04134107516515135099198813769, −5.97784527345172530967753789503, −5.06421808955392790753205419238, −4.92632358264519517447241518087, −4.16593155706803711257900199501, −4.07509436663480773299586041954, −3.43448622087503449119383155452, −3.41947163587125376660483214850, −2.67529815770567707012049907024, −2.62281502463820647787683475795, −1.50970162287548745186148872109, −1.48670746216841464543140866380, 0, 0,
1.48670746216841464543140866380, 1.50970162287548745186148872109, 2.62281502463820647787683475795, 2.67529815770567707012049907024, 3.41947163587125376660483214850, 3.43448622087503449119383155452, 4.07509436663480773299586041954, 4.16593155706803711257900199501, 4.92632358264519517447241518087, 5.06421808955392790753205419238, 5.97784527345172530967753789503, 6.04134107516515135099198813769, 6.72299191914191127814883950879, 6.87746432065015431590065997132, 7.26267370186674390213852162645, 7.48206244061132668655557186412, 8.105500556012422632035757697208, 8.162244655544744862092529867049
