L(s) = 1 | − 2·3-s − 4·5-s + 9-s + 2·11-s − 2·13-s + 8·15-s + 5·17-s + 2·19-s − 23-s + 11·25-s + 4·27-s + 2·29-s − 4·33-s − 4·37-s + 4·39-s + 7·41-s − 4·45-s − 4·47-s − 10·51-s − 8·55-s − 4·57-s + 4·59-s − 8·61-s + 8·65-s − 6·67-s + 2·69-s + 4·71-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1.78·5-s + 1/3·9-s + 0.603·11-s − 0.554·13-s + 2.06·15-s + 1.21·17-s + 0.458·19-s − 0.208·23-s + 11/5·25-s + 0.769·27-s + 0.371·29-s − 0.696·33-s − 0.657·37-s + 0.640·39-s + 1.09·41-s − 0.596·45-s − 0.583·47-s − 1.40·51-s − 1.07·55-s − 0.529·57-s + 0.520·59-s − 1.02·61-s + 0.992·65-s − 0.733·67-s + 0.240·69-s + 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4635735848\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4635735848\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 31 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−12.28965130440721, −12.06020397610330, −11.66755306410294, −11.23911296793554, −10.83650313587684, −10.40302948523803, −9.897792720968140, −9.303896816803780, −8.818106512872610, −8.264685852361496, −7.761202074445958, −7.507409505368888, −6.990621368458395, −6.455063581171291, −6.069397770406451, −5.280224832854949, −5.125287615163239, −4.447052459813718, −4.088532206117701, −3.442087601782753, −3.119175843889588, −2.403328332705288, −1.327925655798353, −0.9486767472265419, −0.2453853157897943,
0.2453853157897943, 0.9486767472265419, 1.327925655798353, 2.403328332705288, 3.119175843889588, 3.442087601782753, 4.088532206117701, 4.447052459813718, 5.125287615163239, 5.280224832854949, 6.069397770406451, 6.455063581171291, 6.990621368458395, 7.507409505368888, 7.761202074445958, 8.264685852361496, 8.818106512872610, 9.303896816803780, 9.897792720968140, 10.40302948523803, 10.83650313587684, 11.23911296793554, 11.66755306410294, 12.06020397610330, 12.28965130440721