| L(s) = 1 | − 3-s + 2.56·5-s − 7-s + 9-s − 11-s − 0.561·13-s − 2.56·15-s − 5.12·17-s − 4.56·19-s + 21-s − 4·23-s + 1.56·25-s − 27-s − 0.561·29-s + 2·31-s + 33-s − 2.56·35-s − 4.56·37-s + 0.561·39-s + 10.2·41-s − 10.2·43-s + 2.56·45-s − 13.6·47-s + 49-s + 5.12·51-s + 3.12·53-s − 2.56·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.14·5-s − 0.377·7-s + 0.333·9-s − 0.301·11-s − 0.155·13-s − 0.661·15-s − 1.24·17-s − 1.04·19-s + 0.218·21-s − 0.834·23-s + 0.312·25-s − 0.192·27-s − 0.104·29-s + 0.359·31-s + 0.174·33-s − 0.432·35-s − 0.749·37-s + 0.0899·39-s + 1.60·41-s − 1.56·43-s + 0.381·45-s − 1.99·47-s + 0.142·49-s + 0.717·51-s + 0.428·53-s − 0.345·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 2.56T + 5T^{2} \) |
| 13 | \( 1 + 0.561T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 + 4.56T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 0.561T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 4.56T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 13.6T + 47T^{2} \) |
| 53 | \( 1 - 3.12T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 2.56T + 67T^{2} \) |
| 71 | \( 1 - 2.24T + 71T^{2} \) |
| 73 | \( 1 - 3.68T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 - 7.12T + 97T^{2} \) |
| 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
−8.919954766750834372448795973539, −8.185426373712963037404097378663, −6.95249674622891741865506578159, −6.40910039168689416328769878945, −5.73710391289535026987851271149, −4.88430955898731225087180162183, −3.96547637377342995643313940462, −2.56079219372947162612990446725, −1.75489374438977001622753267559, 0,
1.75489374438977001622753267559, 2.56079219372947162612990446725, 3.96547637377342995643313940462, 4.88430955898731225087180162183, 5.73710391289535026987851271149, 6.40910039168689416328769878945, 6.95249674622891741865506578159, 8.185426373712963037404097378663, 8.919954766750834372448795973539
