L(s) = 1 | + 4·11-s − 2·13-s − 2·17-s + 4·19-s + 10·29-s − 6·37-s − 6·41-s + 4·43-s + 8·47-s + 6·53-s − 4·59-s + 10·61-s − 4·67-s + 16·71-s − 14·73-s + 8·79-s + 4·83-s + 10·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | + 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 1.85·29-s − 0.986·37-s − 0.937·41-s + 0.609·43-s + 1.16·47-s + 0.824·53-s − 0.520·59-s + 1.28·61-s − 0.488·67-s + 1.89·71-s − 1.63·73-s + 0.900·79-s + 0.439·83-s + 1.05·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.947079008\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.947079008\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.04634040805657, −13.48561624077014, −12.90563034723716, −12.22752422784929, −11.81473429341250, −11.76435084982599, −10.81566493354142, −10.44503272452243, −9.896693199493581, −9.357316005741389, −8.896467230192020, −8.467333829399405, −7.808467983869082, −7.176354908199973, −6.777844087687453, −6.323724668791089, −5.626557600395571, −5.022973121157508, −4.544089188243603, −3.876171522833680, −3.366884198767243, −2.635679100142959, −2.022523592086488, −1.193775977149467, −0.6068972067051589,
0.6068972067051589, 1.193775977149467, 2.022523592086488, 2.635679100142959, 3.366884198767243, 3.876171522833680, 4.544089188243603, 5.022973121157508, 5.626557600395571, 6.323724668791089, 6.777844087687453, 7.176354908199973, 7.808467983869082, 8.467333829399405, 8.896467230192020, 9.357316005741389, 9.896693199493581, 10.44503272452243, 10.81566493354142, 11.76435084982599, 11.81473429341250, 12.22752422784929, 12.90563034723716, 13.48561624077014, 14.04634040805657