| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 4·6-s + 4·8-s + 3·9-s + 4·11-s − 6·12-s − 6·13-s + 5·16-s − 6·17-s + 6·18-s + 8·22-s − 2·23-s − 8·24-s − 12·26-s − 4·27-s + 2·29-s − 2·31-s + 6·32-s − 8·33-s − 12·34-s + 9·36-s + 12·39-s − 6·41-s − 6·43-s + 12·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.63·6-s + 1.41·8-s + 9-s + 1.20·11-s − 1.73·12-s − 1.66·13-s + 5/4·16-s − 1.45·17-s + 1.41·18-s + 1.70·22-s − 0.417·23-s − 1.63·24-s − 2.35·26-s − 0.769·27-s + 0.371·29-s − 0.359·31-s + 1.06·32-s − 1.39·33-s − 2.05·34-s + 3/2·36-s + 1.92·39-s − 0.937·41-s − 0.914·43-s + 1.80·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54022500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54022500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 11 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 45 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 61 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 93 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 109 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 123 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 160 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 144 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 165 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.44405355316736223063804502174, −7.15682575473611165167901913667, −6.82173983155070561746793595256, −6.63751982469528925412081091255, −6.19242201192679074539163602985, −6.17748654935949412502055366768, −5.38434481456279028914926181174, −5.33039647286723210289858331931, −4.75716456657775300532137445232, −4.71719259190885492050295794517, −4.15813146499955283073291242915, −4.09225220054830573058358523438, −3.30010295712569470546693104402, −3.20551150068266142717276896923, −2.34858306802232441305385644393, −2.26291115586901085064660806039, −1.42986845095495324780145435791, −1.38815644151582075873748481269, 0, 0,
1.38815644151582075873748481269, 1.42986845095495324780145435791, 2.26291115586901085064660806039, 2.34858306802232441305385644393, 3.20551150068266142717276896923, 3.30010295712569470546693104402, 4.09225220054830573058358523438, 4.15813146499955283073291242915, 4.71719259190885492050295794517, 4.75716456657775300532137445232, 5.33039647286723210289858331931, 5.38434481456279028914926181174, 6.17748654935949412502055366768, 6.19242201192679074539163602985, 6.63751982469528925412081091255, 6.82173983155070561746793595256, 7.15682575473611165167901913667, 7.44405355316736223063804502174
