| L(s) = 1 | − 1.66·5-s − 3.57i·7-s + 1.02·11-s + (−1.87 − 3.07i)13-s − 5.05·17-s + 1.16·19-s − 8.17·23-s − 2.23·25-s − 4.29i·29-s + 7.98i·31-s + 5.93i·35-s + 9.83·37-s − 1.62i·41-s + 2.35i·43-s + 7.73i·47-s + ⋯ |
| L(s) = 1 | − 0.743·5-s − 1.35i·7-s + 0.309·11-s + (−0.520 − 0.853i)13-s − 1.22·17-s + 0.266·19-s − 1.70·23-s − 0.447·25-s − 0.798i·29-s + 1.43i·31-s + 1.00i·35-s + 1.61·37-s − 0.253i·41-s + 0.358i·43-s + 1.12i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.231 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2928510183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2928510183\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (1.87 + 3.07i)T \) |
| good | 5 | \( 1 + 1.66T + 5T^{2} \) |
| 7 | \( 1 + 3.57iT - 7T^{2} \) |
| 11 | \( 1 - 1.02T + 11T^{2} \) |
| 17 | \( 1 + 5.05T + 17T^{2} \) |
| 19 | \( 1 - 1.16T + 19T^{2} \) |
| 23 | \( 1 + 8.17T + 23T^{2} \) |
| 29 | \( 1 + 4.29iT - 29T^{2} \) |
| 31 | \( 1 - 7.98iT - 31T^{2} \) |
| 37 | \( 1 - 9.83T + 37T^{2} \) |
| 41 | \( 1 + 1.62iT - 41T^{2} \) |
| 43 | \( 1 - 2.35iT - 43T^{2} \) |
| 47 | \( 1 - 7.73iT - 47T^{2} \) |
| 53 | \( 1 - 11.2iT - 53T^{2} \) |
| 59 | \( 1 - 9.73T + 59T^{2} \) |
| 61 | \( 1 + 11.4iT - 61T^{2} \) |
| 67 | \( 1 + 7.23T + 67T^{2} \) |
| 71 | \( 1 - 10.5iT - 71T^{2} \) |
| 73 | \( 1 + 4.41iT - 73T^{2} \) |
| 79 | \( 1 - 6.94T + 79T^{2} \) |
| 83 | \( 1 - 2.29T + 83T^{2} \) |
| 89 | \( 1 - 9.02iT - 89T^{2} \) |
| 97 | \( 1 + 11.5iT - 97T^{2} \) |
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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
−8.537896114307573217728566255914, −7.79098812869012517568830846865, −7.47983850797115059783418118714, −6.58625311510430993697973253722, −5.85704497643038250882720067793, −4.62046308521651940029600119285, −4.20052813319229153477112545510, −3.43909376981696124980487830632, −2.32970290292608137012075445937, −0.966660467320246193919145738151,
0.099028767953143538141606025447, 1.96720616184811833589236402002, 2.49882721629691358955915111810, 3.81916910212406111405096955914, 4.34235161437424898306707341940, 5.32540919461722595189099181861, 6.11053045071106354504893125227, 6.76240251237239415525371106984, 7.69923203910681693295298309543, 8.288975592916366502735754319971
