| L(s) = 1 | + 2-s + 4-s + 1.95·5-s + 0.458·7-s + 8-s + 1.95·10-s + 1.25·11-s + 5.60·13-s + 0.458·14-s + 16-s + 6.54·17-s − 2.11·19-s + 1.95·20-s + 1.25·22-s − 1.19·25-s + 5.60·26-s + 0.458·28-s − 5.06·29-s − 0.109·31-s + 32-s + 6.54·34-s + 0.894·35-s + 0.926·37-s − 2.11·38-s + 1.95·40-s + 9.83·41-s − 6.77·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.872·5-s + 0.173·7-s + 0.353·8-s + 0.617·10-s + 0.379·11-s + 1.55·13-s + 0.122·14-s + 0.250·16-s + 1.58·17-s − 0.486·19-s + 0.436·20-s + 0.268·22-s − 0.238·25-s + 1.09·26-s + 0.0865·28-s − 0.940·29-s − 0.0195·31-s + 0.176·32-s + 1.12·34-s + 0.151·35-s + 0.152·37-s − 0.343·38-s + 0.308·40-s + 1.53·41-s − 1.03·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.086185144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.086185144\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 1.95T + 5T^{2} \) |
| 7 | \( 1 - 0.458T + 7T^{2} \) |
| 11 | \( 1 - 1.25T + 11T^{2} \) |
| 13 | \( 1 - 5.60T + 13T^{2} \) |
| 17 | \( 1 - 6.54T + 17T^{2} \) |
| 19 | \( 1 + 2.11T + 19T^{2} \) |
| 29 | \( 1 + 5.06T + 29T^{2} \) |
| 31 | \( 1 + 0.109T + 31T^{2} \) |
| 37 | \( 1 - 0.926T + 37T^{2} \) |
| 41 | \( 1 - 9.83T + 41T^{2} \) |
| 43 | \( 1 + 6.77T + 43T^{2} \) |
| 47 | \( 1 - 2.50T + 47T^{2} \) |
| 53 | \( 1 + 2.64T + 53T^{2} \) |
| 59 | \( 1 - 4.27T + 59T^{2} \) |
| 61 | \( 1 - 8.58T + 61T^{2} \) |
| 67 | \( 1 - 15.0T + 67T^{2} \) |
| 71 | \( 1 - 0.303T + 71T^{2} \) |
| 73 | \( 1 - 3.34T + 73T^{2} \) |
| 79 | \( 1 - 8.04T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 5.95T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
| 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
−7.77689828858812954970470916181, −6.65045489876445725278803436208, −6.30657814184972563173063643197, −5.52776559295767085754296127030, −5.22249405308544633323123169456, −3.93028033199425892945423166973, −3.70521204113065743122095197303, −2.65190681867174797032010443056, −1.75110601686761748056477131991, −1.06295624055603552724344967648,
1.06295624055603552724344967648, 1.75110601686761748056477131991, 2.65190681867174797032010443056, 3.70521204113065743122095197303, 3.93028033199425892945423166973, 5.22249405308544633323123169456, 5.52776559295767085754296127030, 6.30657814184972563173063643197, 6.65045489876445725278803436208, 7.77689828858812954970470916181
