| L(s) = 1 | − 2·3-s + 4·7-s + 3·9-s − 8·19-s − 8·21-s + 10·25-s − 4·27-s + 12·29-s − 8·31-s + 4·37-s + 9·49-s + 12·53-s + 16·57-s + 24·59-s + 12·63-s − 20·75-s + 5·81-s + 24·83-s − 24·87-s + 16·93-s − 8·103-s + 20·109-s − 8·111-s − 12·113-s + 10·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.51·7-s + 9-s − 1.83·19-s − 1.74·21-s + 2·25-s − 0.769·27-s + 2.22·29-s − 1.43·31-s + 0.657·37-s + 9/7·49-s + 1.64·53-s + 2.11·57-s + 3.12·59-s + 1.51·63-s − 2.30·75-s + 5/9·81-s + 2.63·83-s − 2.57·87-s + 1.65·93-s − 0.788·103-s + 1.91·109-s − 0.759·111-s − 1.12·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.941408526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.941408526\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−10.06995724220776190299800585438, −9.378516593386338853913124990283, −8.775035843017275971164114927853, −8.709007801533398758380195632651, −8.158916265086921843487717291340, −7.941659312566472200500565493962, −7.09620859889848851217462396786, −7.00595061998234371103301050381, −6.54985990570191923835023820910, −6.12911574601779247151672322914, −5.47560063473087330767393352362, −5.27077084651351830351565786783, −4.69055590738666400724028010554, −4.50101164573175574587615596859, −4.01327240451662973259573863110, −3.34371916209221644052043072861, −2.30826990752695288933640422042, −2.22237076072878060709891239409, −1.18242960898448693491420044661, −0.72816441269321262951392421027,
0.72816441269321262951392421027, 1.18242960898448693491420044661, 2.22237076072878060709891239409, 2.30826990752695288933640422042, 3.34371916209221644052043072861, 4.01327240451662973259573863110, 4.50101164573175574587615596859, 4.69055590738666400724028010554, 5.27077084651351830351565786783, 5.47560063473087330767393352362, 6.12911574601779247151672322914, 6.54985990570191923835023820910, 7.00595061998234371103301050381, 7.09620859889848851217462396786, 7.941659312566472200500565493962, 8.158916265086921843487717291340, 8.709007801533398758380195632651, 8.775035843017275971164114927853, 9.378516593386338853913124990283, 10.06995724220776190299800585438
