| L(s) = 1 | − 3-s + 4-s − 5-s + 3·7-s − 9-s − 11-s − 12-s − 6·13-s + 15-s + 16-s − 2·17-s − 19-s − 20-s − 3·21-s + 23-s + 5·25-s + 3·28-s − 4·29-s − 10·31-s + 33-s − 3·35-s − 36-s + 6·39-s − 5·41-s − 8·43-s − 44-s + 45-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 0.447·5-s + 1.13·7-s − 1/3·9-s − 0.301·11-s − 0.288·12-s − 1.66·13-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.229·19-s − 0.223·20-s − 0.654·21-s + 0.208·23-s + 25-s + 0.566·28-s − 0.742·29-s − 1.79·31-s + 0.174·33-s − 0.507·35-s − 1/6·36-s + 0.960·39-s − 0.780·41-s − 1.21·43-s − 0.150·44-s + 0.149·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100124 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100124 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 25031 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 56 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 6 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - T + 14 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 5 T + 60 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 11 T + 96 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T - 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−14.2370415270, −14.1480268781, −13.1910040667, −12.9219795164, −12.3940342810, −12.0167818111, −11.5580397260, −11.2257067140, −10.8889237244, −10.4479575210, −9.92964136691, −9.22295989435, −8.88437776350, −8.21611952188, −7.73083057298, −7.37120760139, −6.89720051130, −6.33949772657, −5.46806105891, −5.20316792719, −4.75225418138, −4.02226831550, −3.17650644070, −2.37035152202, −1.68203006173, 0,
1.68203006173, 2.37035152202, 3.17650644070, 4.02226831550, 4.75225418138, 5.20316792719, 5.46806105891, 6.33949772657, 6.89720051130, 7.37120760139, 7.73083057298, 8.21611952188, 8.88437776350, 9.22295989435, 9.92964136691, 10.4479575210, 10.8889237244, 11.2257067140, 11.5580397260, 12.0167818111, 12.3940342810, 12.9219795164, 13.1910040667, 14.1480268781, 14.2370415270
