L(s) = 1 | + (1.68 − 0.417i)3-s − 5-s + 2.42i·7-s + (2.65 − 1.40i)9-s + 2.52·11-s + 0.420i·13-s + (−1.68 + 0.417i)15-s + 1.53i·17-s − 1.99·19-s + (1.01 + 4.08i)21-s + 2.90i·23-s + 25-s + (3.87 − 3.46i)27-s + 1.32i·29-s − 4.77i·31-s + ⋯ |
L(s) = 1 | + (0.970 − 0.240i)3-s − 0.447·5-s + 0.917i·7-s + (0.883 − 0.467i)9-s + 0.762·11-s + 0.116i·13-s + (−0.434 + 0.107i)15-s + 0.371i·17-s − 0.456·19-s + (0.221 + 0.890i)21-s + 0.606i·23-s + 0.200·25-s + (0.745 − 0.666i)27-s + 0.245i·29-s − 0.856i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.646393507\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.646393507\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.68 + 0.417i)T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (-7.34 + 3.61i)T \) |
good | 7 | \( 1 - 2.42iT - 7T^{2} \) |
| 11 | \( 1 - 2.52T + 11T^{2} \) |
| 13 | \( 1 - 0.420iT - 13T^{2} \) |
| 17 | \( 1 - 1.53iT - 17T^{2} \) |
| 19 | \( 1 + 1.99T + 19T^{2} \) |
| 23 | \( 1 - 2.90iT - 23T^{2} \) |
| 29 | \( 1 - 1.32iT - 29T^{2} \) |
| 31 | \( 1 + 4.77iT - 31T^{2} \) |
| 37 | \( 1 - 9.45T + 37T^{2} \) |
| 41 | \( 1 + 2.00T + 41T^{2} \) |
| 43 | \( 1 - 2.63iT - 43T^{2} \) |
| 47 | \( 1 - 2.60iT - 47T^{2} \) |
| 53 | \( 1 + 1.37T + 53T^{2} \) |
| 59 | \( 1 - 14.2iT - 59T^{2} \) |
| 61 | \( 1 - 11.2iT - 61T^{2} \) |
| 71 | \( 1 - 3.60iT - 71T^{2} \) |
| 73 | \( 1 + 1.92T + 73T^{2} \) |
| 79 | \( 1 - 7.44iT - 79T^{2} \) |
| 83 | \( 1 + 13.4iT - 83T^{2} \) |
| 89 | \( 1 + 2.46iT - 89T^{2} \) |
| 97 | \( 1 - 4.51iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.572312932428567873844775392493, −7.87634093011151687552743330390, −7.22482009972085211878313785539, −6.36603010211350567596505478228, −5.71491862712358306930924017869, −4.47182289289130548756741965246, −3.92378164858048611056336303473, −2.97029975150945564233094154748, −2.20761539874600550994216250489, −1.17115960304260652645019318615,
0.74560994039348872467318853073, 1.92987328122089662877498856229, 3.00294206345096525799139552864, 3.79821669846159529703485310198, 4.33022976522561696984971702890, 5.10943604623212498701542223100, 6.48940528512651828788905264581, 6.94045803008036794551779300870, 7.80597477828599615988849764304, 8.259633174329648935726085724585