L(s) = 1 | + 2·2-s − 4-s − 8·8-s − 6·9-s − 6·11-s − 7·16-s − 12·18-s − 12·22-s − 12·23-s − 10·25-s − 10·29-s + 14·32-s + 6·36-s + 16·37-s + 4·43-s + 6·44-s − 24·46-s − 20·50-s − 6·53-s − 20·58-s + 35·64-s − 6·67-s − 10·71-s + 48·72-s + 32·74-s − 12·79-s + 27·81-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 1/2·4-s − 2.82·8-s − 2·9-s − 1.80·11-s − 7/4·16-s − 2.82·18-s − 2.55·22-s − 2.50·23-s − 2·25-s − 1.85·29-s + 2.47·32-s + 36-s + 2.63·37-s + 0.609·43-s + 0.904·44-s − 3.53·46-s − 2.82·50-s − 0.824·53-s − 2.62·58-s + 35/8·64-s − 0.733·67-s − 1.18·71-s + 5.65·72-s + 3.71·74-s − 1.35·79-s + 3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 7 | | \( 1 \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.264528127864852069730211246497, −7.71886042427351779334180674638, −7.58221538537525761723731254609, −6.13619288958387853257590270753, −5.97764942510297289458867575451, −5.71244157014039939594521038014, −5.56220221892590651265131476874, −4.57403153778540150751986491981, −4.53493125422219918903828313314, −3.54025539217060415386292648543, −3.52729931218747647603184716659, −2.43258577292117985821674468072, −2.43005796671753573111852467369, 0, 0,
2.43005796671753573111852467369, 2.43258577292117985821674468072, 3.52729931218747647603184716659, 3.54025539217060415386292648543, 4.53493125422219918903828313314, 4.57403153778540150751986491981, 5.56220221892590651265131476874, 5.71244157014039939594521038014, 5.97764942510297289458867575451, 6.13619288958387853257590270753, 7.58221538537525761723731254609, 7.71886042427351779334180674638, 8.264528127864852069730211246497