| L(s) = 1 | + 2.65i·3-s − 5-s + 1.97i·7-s − 4.04·9-s − 5.81i·11-s − 1.97i·13-s − 2.65i·15-s + 2.65i·17-s − 8.26i·19-s − 5.23·21-s − 3.04·23-s + 25-s − 2.76i·27-s − 7.39i·29-s − 6.49·31-s + ⋯ |
| L(s) = 1 | + 1.53i·3-s − 0.447·5-s + 0.745i·7-s − 1.34·9-s − 1.75i·11-s − 0.546i·13-s − 0.685i·15-s + 0.643i·17-s − 1.89i·19-s − 1.14·21-s − 0.634·23-s + 0.200·25-s − 0.532i·27-s − 1.37i·29-s − 1.16·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7682168435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7682168435\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 41 | \( 1 + (4.30 + 4.74i)T \) |
| good | 3 | \( 1 - 2.65iT - 3T^{2} \) |
| 7 | \( 1 - 1.97iT - 7T^{2} \) |
| 11 | \( 1 + 5.81iT - 11T^{2} \) |
| 13 | \( 1 + 1.97iT - 13T^{2} \) |
| 17 | \( 1 - 2.65iT - 17T^{2} \) |
| 19 | \( 1 + 8.26iT - 19T^{2} \) |
| 23 | \( 1 + 3.04T + 23T^{2} \) |
| 29 | \( 1 + 7.39iT - 29T^{2} \) |
| 31 | \( 1 + 6.49T + 31T^{2} \) |
| 37 | \( 1 - 1.45T + 37T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 10.0iT - 47T^{2} \) |
| 53 | \( 1 - 9.33iT - 53T^{2} \) |
| 59 | \( 1 - 7.15T + 59T^{2} \) |
| 61 | \( 1 - 9.72T + 61T^{2} \) |
| 67 | \( 1 - 1.15iT - 67T^{2} \) |
| 71 | \( 1 - 4.05iT - 71T^{2} \) |
| 73 | \( 1 + 1.59T + 73T^{2} \) |
| 79 | \( 1 - 1.57iT - 79T^{2} \) |
| 83 | \( 1 - 4.11T + 83T^{2} \) |
| 89 | \( 1 + 17.9iT - 89T^{2} \) |
| 97 | \( 1 - 12.8iT - 97T^{2} \) |
| 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
−9.139971828492265564348647188568, −8.687358924920554430164754236628, −8.071793308220773874806007203822, −6.72959813393347939876287851724, −5.64403730222312693147208662376, −5.26000144813132876462206314397, −4.09745345521367322470836044310, −3.46399976711725860526999046659, −2.56015149172486944881949137532, −0.29619079096306906098490376080,
1.40375150233782228300218548901, 2.03546466560503920474742035875, 3.49983361075254148535155330038, 4.45225240480219054903571784719, 5.51015308869037168052884429165, 6.82357110662940211354406547995, 6.93575743057629344395532924264, 7.78083026645325736212266234364, 8.290933198029373123355228148285, 9.558536054423451599104148525407
