| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s − 5·11-s − 2·12-s + 16-s − 7·17-s − 18-s + 13·19-s + 5·22-s + 2·24-s + 2·25-s − 2·27-s − 32-s + 10·33-s + 7·34-s + 36-s − 13·38-s − 6·41-s − 3·43-s − 5·44-s − 2·48-s + 8·49-s − 2·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 1.50·11-s − 0.577·12-s + 1/4·16-s − 1.69·17-s − 0.235·18-s + 2.98·19-s + 1.06·22-s + 0.408·24-s + 2/5·25-s − 0.384·27-s − 0.176·32-s + 1.74·33-s + 1.20·34-s + 1/6·36-s − 2.10·38-s − 0.937·41-s − 0.457·43-s − 0.753·44-s − 0.288·48-s + 8/7·49-s − 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 4003 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 47 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 45 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 39 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 112 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 51 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.218370766515906982668323145289, −7.72905440451200409755776381744, −7.37958624138402390706585863494, −7.02059764161955810083641382272, −6.45433881005875128622935282094, −5.95543963637370219124967206822, −5.42945589889596653402409459654, −5.13716135656228276479200650833, −4.78367113571855640759689509783, −3.88689663136369299444641662582, −3.15964128625002365457527727412, −2.67331264238037879036134903043, −1.90287926746018353490855373048, −0.906985092150766730586953539089, 0,
0.906985092150766730586953539089, 1.90287926746018353490855373048, 2.67331264238037879036134903043, 3.15964128625002365457527727412, 3.88689663136369299444641662582, 4.78367113571855640759689509783, 5.13716135656228276479200650833, 5.42945589889596653402409459654, 5.95543963637370219124967206822, 6.45433881005875128622935282094, 7.02059764161955810083641382272, 7.37958624138402390706585863494, 7.72905440451200409755776381744, 8.218370766515906982668323145289
