| L(s) = 1 | + 2·5-s + 3.79·7-s − 11-s + 13-s − 8.37·19-s − 0.208·23-s − 25-s − 7.58·29-s + 4·31-s + 7.58·35-s + 8·37-s + 5.79·41-s + 8·43-s + 11.5·47-s + 7.37·49-s + 9.95·53-s − 2·55-s + 9.58·59-s + 5.58·61-s + 2·65-s + 8·67-s + 8·71-s − 12.3·73-s − 3.79·77-s + 2.41·79-s − 3.37·83-s + 13.5·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.43·7-s − 0.301·11-s + 0.277·13-s − 1.92·19-s − 0.0435·23-s − 0.200·25-s − 1.40·29-s + 0.718·31-s + 1.28·35-s + 1.31·37-s + 0.904·41-s + 1.21·43-s + 1.68·47-s + 1.05·49-s + 1.36·53-s − 0.269·55-s + 1.24·59-s + 0.714·61-s + 0.248·65-s + 0.977·67-s + 0.949·71-s − 1.44·73-s − 0.432·77-s + 0.271·79-s − 0.370·83-s + 1.43·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.784946294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.784946294\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 - 3.79T + 7T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 8.37T + 19T^{2} \) |
| 23 | \( 1 + 0.208T + 23T^{2} \) |
| 29 | \( 1 + 7.58T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 5.79T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 9.95T + 53T^{2} \) |
| 59 | \( 1 - 9.58T + 59T^{2} \) |
| 61 | \( 1 - 5.58T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 2.41T + 79T^{2} \) |
| 83 | \( 1 + 3.37T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
| 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
−8.228556548980868738412043191629, −7.62404590366383608526705941239, −6.76190445469600947358099620310, −5.79310688912870174246221035532, −5.55395845652764328894823819924, −4.41710072675034944335081320088, −4.02503396655168249846568095302, −2.37383749268971025559944515226, −2.12512339701794802893176341293, −0.933730323220071320825337742520,
0.933730323220071320825337742520, 2.12512339701794802893176341293, 2.37383749268971025559944515226, 4.02503396655168249846568095302, 4.41710072675034944335081320088, 5.55395845652764328894823819924, 5.79310688912870174246221035532, 6.76190445469600947358099620310, 7.62404590366383608526705941239, 8.228556548980868738412043191629
