L(s) = 1 | − 4-s + 7-s + 9·13-s + 16-s + 8·25-s − 28-s + 5·37-s − 7·43-s − 6·49-s − 9·52-s + 9·61-s − 64-s − 17·67-s − 9·73-s − 11·79-s + 9·91-s − 8·100-s + 14·109-s + 112-s + 4·121-s + 127-s + 131-s + 137-s + 139-s − 5·148-s + 149-s + 151-s + ⋯ |
L(s) = 1 | − 1/2·4-s + 0.377·7-s + 2.49·13-s + 1/4·16-s + 8/5·25-s − 0.188·28-s + 0.821·37-s − 1.06·43-s − 6/7·49-s − 1.24·52-s + 1.15·61-s − 1/8·64-s − 2.07·67-s − 1.05·73-s − 1.23·79-s + 0.943·91-s − 4/5·100-s + 1.34·109-s + 0.0944·112-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.410·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.729774751\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.729774751\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.139653344749279964318027512398, −8.752773473131153470061873036103, −8.460607111207407981112349907621, −8.076683354683520637597902581608, −7.40572156925869950955407390609, −6.78347510071548608962964563042, −6.31331610173668692333696289993, −5.82067067992331099005958496359, −5.31163955771378476686658667715, −4.53748311300456991048958963030, −4.22307191307728643314985931790, −3.37658620712098305101864082824, −3.01922594705373137377702138338, −1.72715717203838961223003097116, −1.01441521593897187512941634728,
1.01441521593897187512941634728, 1.72715717203838961223003097116, 3.01922594705373137377702138338, 3.37658620712098305101864082824, 4.22307191307728643314985931790, 4.53748311300456991048958963030, 5.31163955771378476686658667715, 5.82067067992331099005958496359, 6.31331610173668692333696289993, 6.78347510071548608962964563042, 7.40572156925869950955407390609, 8.076683354683520637597902581608, 8.460607111207407981112349907621, 8.752773473131153470061873036103, 9.139653344749279964318027512398