L(s) = 1 | + 2·2-s + 2·4-s + 5-s − 2·7-s + 2·10-s + 3·11-s − 5·13-s − 4·14-s − 4·16-s + 17-s − 19-s + 2·20-s + 6·22-s + 7·23-s − 4·25-s − 10·26-s − 4·28-s − 6·29-s + 4·31-s − 8·32-s + 2·34-s − 2·35-s + 10·37-s − 2·38-s − 9·41-s + 43-s + 6·44-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s + 0.447·5-s − 0.755·7-s + 0.632·10-s + 0.904·11-s − 1.38·13-s − 1.06·14-s − 16-s + 0.242·17-s − 0.229·19-s + 0.447·20-s + 1.27·22-s + 1.45·23-s − 4/5·25-s − 1.96·26-s − 0.755·28-s − 1.11·29-s + 0.718·31-s − 1.41·32-s + 0.342·34-s − 0.338·35-s + 1.64·37-s − 0.324·38-s − 1.40·41-s + 0.152·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.076797451\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.076797451\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.04673624974979209240063543467, −12.28350396167249658664748486713, −11.40917261844359191960924897433, −9.902853599394832668339931927447, −9.106190491838796506296814247251, −7.21604121649276372487152079314, −6.25462947962542131992450139682, −5.19812353125681781678502892557, −3.95590573967765265392229268688, −2.62397813274243291040838700940,
2.62397813274243291040838700940, 3.95590573967765265392229268688, 5.19812353125681781678502892557, 6.25462947962542131992450139682, 7.21604121649276372487152079314, 9.106190491838796506296814247251, 9.902853599394832668339931927447, 11.40917261844359191960924897433, 12.28350396167249658664748486713, 13.04673624974979209240063543467