| L(s) = 1 | − 2·2-s − 2·3-s + 4-s + 4·6-s − 2·7-s + 3·9-s − 4·11-s − 2·12-s + 4·14-s + 16-s + 4·17-s − 6·18-s + 4·21-s + 8·22-s + 8·23-s − 10·25-s − 4·27-s − 2·28-s + 4·29-s + 8·31-s + 2·32-s + 8·33-s − 8·34-s + 3·36-s + 4·37-s − 8·42-s + 8·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s + 1.63·6-s − 0.755·7-s + 9-s − 1.20·11-s − 0.577·12-s + 1.06·14-s + 1/4·16-s + 0.970·17-s − 1.41·18-s + 0.872·21-s + 1.70·22-s + 1.66·23-s − 2·25-s − 0.769·27-s − 0.377·28-s + 0.742·29-s + 1.43·31-s + 0.353·32-s + 1.39·33-s − 1.37·34-s + 1/2·36-s + 0.657·37-s − 1.23·42-s + 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12595401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12595401 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T - 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.264327237365188141958637622305, −8.189778319041715810788406124272, −7.54536604695939682612283274822, −7.36111289784737115780983659547, −7.21101332027950977120221985613, −6.34349483225879552596542727207, −6.06435023580241889068945571360, −6.04567652736480033501320650508, −5.53632582481203793890680934465, −4.87725834449547273895303117685, −4.69032007132824193202278619303, −4.35382642449064262643723392224, −3.52345276384268315018705570323, −3.13377931740056975977858363869, −2.77147271746205523566720233246, −2.15057719051270198870369330596, −1.17899244567508364258277412293, −1.06181339963306821650290580281, 0, 0,
1.06181339963306821650290580281, 1.17899244567508364258277412293, 2.15057719051270198870369330596, 2.77147271746205523566720233246, 3.13377931740056975977858363869, 3.52345276384268315018705570323, 4.35382642449064262643723392224, 4.69032007132824193202278619303, 4.87725834449547273895303117685, 5.53632582481203793890680934465, 6.04567652736480033501320650508, 6.06435023580241889068945571360, 6.34349483225879552596542727207, 7.21101332027950977120221985613, 7.36111289784737115780983659547, 7.54536604695939682612283274822, 8.189778319041715810788406124272, 8.264327237365188141958637622305
