| L(s) = 1 | + (−1.64 − 1.51i)5-s − 2.05·7-s + 5.60·11-s + 2.63i·13-s + 2.12i·17-s − 4.87i·19-s + (−4.47 + 1.71i)23-s + (0.412 + 4.98i)25-s − 1.54i·29-s − 6.05·31-s + (3.38 + 3.11i)35-s − 0.289·37-s − 2.93i·41-s + 12.2·43-s + 0.475·47-s + ⋯ |
| L(s) = 1 | + (−0.735 − 0.677i)5-s − 0.777·7-s + 1.68·11-s + 0.729i·13-s + 0.516i·17-s − 1.11i·19-s + (−0.933 + 0.358i)23-s + (0.0825 + 0.996i)25-s − 0.287i·29-s − 1.08·31-s + (0.571 + 0.526i)35-s − 0.0475·37-s − 0.459i·41-s + 1.86·43-s + 0.0693·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.288836171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.288836171\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.64 + 1.51i)T \) |
| 23 | \( 1 + (4.47 - 1.71i)T \) |
| good | 7 | \( 1 + 2.05T + 7T^{2} \) |
| 11 | \( 1 - 5.60T + 11T^{2} \) |
| 13 | \( 1 - 2.63iT - 13T^{2} \) |
| 17 | \( 1 - 2.12iT - 17T^{2} \) |
| 19 | \( 1 + 4.87iT - 19T^{2} \) |
| 29 | \( 1 + 1.54iT - 29T^{2} \) |
| 31 | \( 1 + 6.05T + 31T^{2} \) |
| 37 | \( 1 + 0.289T + 37T^{2} \) |
| 41 | \( 1 + 2.93iT - 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 - 0.475T + 47T^{2} \) |
| 53 | \( 1 - 11.3iT - 53T^{2} \) |
| 59 | \( 1 - 12.6iT - 59T^{2} \) |
| 61 | \( 1 + 0.760iT - 61T^{2} \) |
| 67 | \( 1 + 0.897T + 67T^{2} \) |
| 71 | \( 1 - 1.57iT - 71T^{2} \) |
| 73 | \( 1 - 3.69iT - 73T^{2} \) |
| 79 | \( 1 + 13.5iT - 79T^{2} \) |
| 83 | \( 1 - 6.38iT - 83T^{2} \) |
| 89 | \( 1 - 4.36T + 89T^{2} \) |
| 97 | \( 1 - 1.66T + 97T^{2} \) |
| 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
−8.761467543615659560868760000224, −7.63805903706063516692302231378, −7.07889173184253464231657485924, −6.30236955889993417788904813153, −5.64756776128107034917433471295, −4.34333833505242885366346363485, −4.11636338877824032472770180090, −3.23853597103720838846548797192, −1.92201436748508551271970027034, −0.853377173575247283790211450951,
0.49456566848104243703416134817, 1.89250246496186369229561122725, 3.12209452103803498295818609136, 3.67892126006383402046356593698, 4.28996299169095250172969164511, 5.54395656266582376492362934836, 6.33943655540851683025960685550, 6.78248614976635079073850352333, 7.62409332695965105471488963865, 8.230589278029962829991098673005
