| L(s) = 1 | + 2·2-s + 2·4-s − 5-s + 3·7-s − 2·10-s + 3·11-s − 6·13-s + 6·14-s − 4·16-s − 3·17-s − 19-s − 2·20-s + 6·22-s − 4·23-s − 4·25-s − 12·26-s + 6·28-s + 10·29-s + 2·31-s − 8·32-s − 6·34-s − 3·35-s + 8·37-s − 2·38-s + 8·41-s − 43-s + 6·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.447·5-s + 1.13·7-s − 0.632·10-s + 0.904·11-s − 1.66·13-s + 1.60·14-s − 16-s − 0.727·17-s − 0.229·19-s − 0.447·20-s + 1.27·22-s − 0.834·23-s − 4/5·25-s − 2.35·26-s + 1.13·28-s + 1.85·29-s + 0.359·31-s − 1.41·32-s − 1.02·34-s − 0.507·35-s + 1.31·37-s − 0.324·38-s + 1.24·41-s − 0.152·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.140174037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.140174037\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.70402288938109990188600775185, −11.84438281704762988883998671737, −11.42285722847365155593370717983, −9.886397115513209246726221997818, −8.513296718878561122779255009178, −7.33179869161333338136909211767, −6.12831923497636436478284774657, −4.75227634922127063518685270995, −4.20883425493377341015603476651, −2.44031011733442394942262196334,
2.44031011733442394942262196334, 4.20883425493377341015603476651, 4.75227634922127063518685270995, 6.12831923497636436478284774657, 7.33179869161333338136909211767, 8.513296718878561122779255009178, 9.886397115513209246726221997818, 11.42285722847365155593370717983, 11.84438281704762988883998671737, 12.70402288938109990188600775185
