| L(s) = 1 | − 1.41·3-s − 1.41·5-s + 2·7-s − 0.999·9-s + 1.41·11-s − 1.41·13-s + 2.00·15-s + 2·17-s − 4.24·19-s − 2.82·21-s + 6·23-s − 2.99·25-s + 5.65·27-s − 4.24·29-s + 8·31-s − 2.00·33-s − 2.82·35-s + 4.24·37-s + 2.00·39-s + 7.07·43-s + 1.41·45-s + 8·47-s − 3·49-s − 2.82·51-s + 7.07·53-s − 2.00·55-s + 6·57-s + ⋯ |
| L(s) = 1 | − 0.816·3-s − 0.632·5-s + 0.755·7-s − 0.333·9-s + 0.426·11-s − 0.392·13-s + 0.516·15-s + 0.485·17-s − 0.973·19-s − 0.617·21-s + 1.25·23-s − 0.599·25-s + 1.08·27-s − 0.787·29-s + 1.43·31-s − 0.348·33-s − 0.478·35-s + 0.697·37-s + 0.320·39-s + 1.07·43-s + 0.210·45-s + 1.16·47-s − 0.428·49-s − 0.396·51-s + 0.971·53-s − 0.269·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.041190433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.041190433\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 7.07T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 7.07T + 53T^{2} \) |
| 59 | \( 1 - 4.24T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 7.07T + 67T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 1.41T + 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
| 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
−10.11746964753947292584623582371, −9.022895325882427049514167451752, −8.244032673042430475396139725529, −7.44739389036720333325609749594, −6.50387579953710679476530289783, −5.59763915045680867636494472590, −4.76582357539702087439548695148, −3.88620574767162200487105768768, −2.47948089871289773789243062662, −0.829238644358853302781468374646,
0.829238644358853302781468374646, 2.47948089871289773789243062662, 3.88620574767162200487105768768, 4.76582357539702087439548695148, 5.59763915045680867636494472590, 6.50387579953710679476530289783, 7.44739389036720333325609749594, 8.244032673042430475396139725529, 9.022895325882427049514167451752, 10.11746964753947292584623582371
