L(s) = 1 | − 3·3-s + 2·5-s + 2·7-s + 3·9-s + 13-s − 6·15-s − 5·17-s − 2·19-s − 6·21-s − 5·23-s + 2·25-s − 5·29-s + 9·31-s + 4·35-s + 6·37-s − 3·39-s + 41-s + 8·43-s + 6·45-s + 5·47-s − 2·49-s + 15·51-s + 12·53-s + 6·57-s + 8·59-s + 4·61-s + 6·63-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 0.894·5-s + 0.755·7-s + 9-s + 0.277·13-s − 1.54·15-s − 1.21·17-s − 0.458·19-s − 1.30·21-s − 1.04·23-s + 2/5·25-s − 0.928·29-s + 1.61·31-s + 0.676·35-s + 0.986·37-s − 0.480·39-s + 0.156·41-s + 1.21·43-s + 0.894·45-s + 0.729·47-s − 2/7·49-s + 2.10·51-s + 1.64·53-s + 0.794·57-s + 1.04·59-s + 0.512·61-s + 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9040964531\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9040964531\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
good | 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 40 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 9 T + 74 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T - 2 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 50 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 7 T + 132 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 14 T + 194 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−14.0471244928, −13.5631421279, −13.0891319322, −12.7851558532, −12.1171802754, −11.7443815624, −11.4073989772, −11.0110533547, −10.6762714265, −10.1843365156, −9.68760108480, −9.20098716667, −8.46734071027, −8.26334639028, −7.40982852263, −6.88733353607, −6.26523951137, −5.95375306048, −5.61076035786, −5.00569009113, −4.40787850940, −3.94123094142, −2.54812339906, −2.00250294521, −0.776894874491,
0.776894874491, 2.00250294521, 2.54812339906, 3.94123094142, 4.40787850940, 5.00569009113, 5.61076035786, 5.95375306048, 6.26523951137, 6.88733353607, 7.40982852263, 8.26334639028, 8.46734071027, 9.20098716667, 9.68760108480, 10.1843365156, 10.6762714265, 11.0110533547, 11.4073989772, 11.7443815624, 12.1171802754, 12.7851558532, 13.0891319322, 13.5631421279, 14.0471244928