L(s) = 1 | + 2·3-s + 5-s + 9-s − 11-s − 4·13-s + 2·15-s + 4·19-s + 6·23-s + 25-s − 4·27-s + 6·29-s + 8·31-s − 2·33-s − 2·37-s − 8·39-s − 6·41-s + 8·43-s + 45-s + 6·47-s + 6·53-s − 55-s + 8·57-s + 12·59-s + 2·61-s − 4·65-s − 10·67-s + 12·69-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 0.516·15-s + 0.917·19-s + 1.25·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s + 1.43·31-s − 0.348·33-s − 0.328·37-s − 1.28·39-s − 0.937·41-s + 1.21·43-s + 0.149·45-s + 0.875·47-s + 0.824·53-s − 0.134·55-s + 1.05·57-s + 1.56·59-s + 0.256·61-s − 0.496·65-s − 1.22·67-s + 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.004012187\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.004012187\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.47032158889108, −12.76934948475318, −12.29232261117880, −11.94847712706227, −11.28424501710068, −10.79450535202464, −10.09340824552229, −9.855305601614417, −9.440117354210022, −8.764178933257402, −8.570010469837603, −7.918859197029111, −7.522509579331648, −6.861678762157600, −6.680172241012388, −5.602840147425345, −5.426681826543264, −4.740082174750447, −4.245043738843101, −3.472449864506611, −2.927483369876837, −2.563501835500041, −2.149602165413296, −1.203071836314759, −0.6267267583731093,
0.6267267583731093, 1.203071836314759, 2.149602165413296, 2.563501835500041, 2.927483369876837, 3.472449864506611, 4.245043738843101, 4.740082174750447, 5.426681826543264, 5.602840147425345, 6.680172241012388, 6.861678762157600, 7.522509579331648, 7.918859197029111, 8.570010469837603, 8.764178933257402, 9.440117354210022, 9.855305601614417, 10.09340824552229, 10.79450535202464, 11.28424501710068, 11.94847712706227, 12.29232261117880, 12.76934948475318, 13.47032158889108