| L(s) = 1 | + 3-s − 7-s − 2·9-s + 11-s − 13-s − 17-s − 2·19-s − 21-s + 23-s − 5·27-s + 7·29-s − 4·31-s + 33-s − 8·37-s − 39-s − 6·41-s − 8·43-s − 7·47-s + 49-s − 51-s − 4·53-s − 2·57-s + 4·59-s − 10·61-s + 2·63-s − 14·67-s + 69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.301·11-s − 0.277·13-s − 0.242·17-s − 0.458·19-s − 0.218·21-s + 0.208·23-s − 0.962·27-s + 1.29·29-s − 0.718·31-s + 0.174·33-s − 1.31·37-s − 0.160·39-s − 0.937·41-s − 1.21·43-s − 1.02·47-s + 1/7·49-s − 0.140·51-s − 0.549·53-s − 0.264·57-s + 0.520·59-s − 1.28·61-s + 0.251·63-s − 1.71·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.109092753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.109092753\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
| 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
−14.26940508449610, −13.62700724991667, −13.45737943062281, −12.75504574061081, −12.14772721295775, −11.84950910844604, −11.19619788128269, −10.64353383314077, −10.12088591036102, −9.622848029636347, −8.946315511526674, −8.670782601707093, −8.180175302210337, −7.521447483272457, −6.923790595849583, −6.396888589691516, −5.942768940177423, −5.060789489747910, −4.766215828207198, −3.856138606853481, −3.284570655614742, −2.912790897547557, −2.063511395854461, −1.524609993821325, −0.3244364422624879,
0.3244364422624879, 1.524609993821325, 2.063511395854461, 2.912790897547557, 3.284570655614742, 3.856138606853481, 4.766215828207198, 5.060789489747910, 5.942768940177423, 6.396888589691516, 6.923790595849583, 7.521447483272457, 8.180175302210337, 8.670782601707093, 8.946315511526674, 9.622848029636347, 10.12088591036102, 10.64353383314077, 11.19619788128269, 11.84950910844604, 12.14772721295775, 12.75504574061081, 13.45737943062281, 13.62700724991667, 14.26940508449610
