L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s − 11-s + 15-s + 17-s + 19-s + 2·21-s + 25-s − 27-s − 8·29-s + 9·31-s + 33-s + 2·35-s − 37-s + 43-s − 45-s − 2·47-s − 3·49-s − 51-s + 6·53-s + 55-s − 57-s − 6·59-s − 7·61-s − 2·63-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.258·15-s + 0.242·17-s + 0.229·19-s + 0.436·21-s + 1/5·25-s − 0.192·27-s − 1.48·29-s + 1.61·31-s + 0.174·33-s + 0.338·35-s − 0.164·37-s + 0.152·43-s − 0.149·45-s − 0.291·47-s − 3/7·49-s − 0.140·51-s + 0.824·53-s + 0.134·55-s − 0.132·57-s − 0.781·59-s − 0.896·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3506807702\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3506807702\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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
−12.54644025585080, −11.91969639713967, −11.88504951760021, −11.14820158741674, −10.83707710498567, −10.27814012363026, −9.904850734969176, −9.453363865891916, −8.966508240803657, −8.444666731301355, −7.742355874881777, −7.624821587126000, −6.899873731348850, −6.578339665011340, −5.950172728518665, −5.666567728064747, −5.006869718438067, −4.543203172113867, −4.019279804125511, −3.443539759603129, −2.979449469120704, −2.423088889271975, −1.590386724635999, −1.031238296022974, −0.1794372206373341,
0.1794372206373341, 1.031238296022974, 1.590386724635999, 2.423088889271975, 2.979449469120704, 3.443539759603129, 4.019279804125511, 4.543203172113867, 5.006869718438067, 5.666567728064747, 5.950172728518665, 6.578339665011340, 6.899873731348850, 7.624821587126000, 7.742355874881777, 8.444666731301355, 8.966508240803657, 9.453363865891916, 9.904850734969176, 10.27814012363026, 10.83707710498567, 11.14820158741674, 11.88504951760021, 11.91969639713967, 12.54644025585080