L(s) = 1 | − 4-s + 4·5-s + 16-s − 8·19-s − 4·20-s + 11·25-s − 14·29-s + 2·31-s − 18·41-s − 49-s − 2·59-s + 6·61-s − 64-s − 6·71-s + 8·76-s + 32·79-s + 4·80-s − 20·89-s − 32·95-s − 11·100-s − 12·101-s + 32·109-s + 14·116-s − 22·121-s − 2·124-s + 24·125-s + 127-s + ⋯ |
L(s) = 1 | − 1/2·4-s + 1.78·5-s + 1/4·16-s − 1.83·19-s − 0.894·20-s + 11/5·25-s − 2.59·29-s + 0.359·31-s − 2.81·41-s − 1/7·49-s − 0.260·59-s + 0.768·61-s − 1/8·64-s − 0.712·71-s + 0.917·76-s + 3.60·79-s + 0.447·80-s − 2.11·89-s − 3.28·95-s − 1.09·100-s − 1.19·101-s + 3.06·109-s + 1.29·116-s − 2·121-s − 0.179·124-s + 2.14·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3572100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3572100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.018070212\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.018070212\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−9.470091773546405513123459425298, −9.149770123829168611879837226981, −8.694362821051594465767223036792, −8.284461468290240471856276836062, −8.093620136275700955652672827060, −7.38594591260685691947817269868, −6.79714984840198387560668406763, −6.72044235117819441967683221610, −6.21313126124882475859604742612, −5.75358463465654804451540163963, −5.32500589981550152037896970706, −5.24436121810505998525943837523, −4.38640417588946615332972826515, −4.28792558143184170364597357479, −3.38247165199816890224337153379, −3.20396870026367943084872760203, −2.23794449331697112902346353363, −1.94400730203552119912087481795, −1.62738591048487181518791342440, −0.49759246455891484856990489852,
0.49759246455891484856990489852, 1.62738591048487181518791342440, 1.94400730203552119912087481795, 2.23794449331697112902346353363, 3.20396870026367943084872760203, 3.38247165199816890224337153379, 4.28792558143184170364597357479, 4.38640417588946615332972826515, 5.24436121810505998525943837523, 5.32500589981550152037896970706, 5.75358463465654804451540163963, 6.21313126124882475859604742612, 6.72044235117819441967683221610, 6.79714984840198387560668406763, 7.38594591260685691947817269868, 8.093620136275700955652672827060, 8.284461468290240471856276836062, 8.694362821051594465767223036792, 9.149770123829168611879837226981, 9.470091773546405513123459425298