| L(s) = 1 | − 3.07i·3-s + 0.328i·5-s − 0.0673·7-s − 6.48·9-s + 3.92·11-s − 3.28·13-s + 1.01·15-s + 6.87·17-s − i·19-s + 0.207i·21-s − 2.75i·23-s + 4.89·25-s + 10.7i·27-s + 6.62·29-s − 8.36i·31-s + ⋯ |
| L(s) = 1 | − 1.77i·3-s + 0.146i·5-s − 0.0254·7-s − 2.16·9-s + 1.18·11-s − 0.910·13-s + 0.261·15-s + 1.66·17-s − 0.229i·19-s + 0.0452i·21-s − 0.574i·23-s + 0.978·25-s + 2.06i·27-s + 1.22·29-s − 1.50i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.947583248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.947583248\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + iT \) |
| 53 | \( 1 + (-5.06 - 5.22i)T \) |
| good | 3 | \( 1 + 3.07iT - 3T^{2} \) |
| 5 | \( 1 - 0.328iT - 5T^{2} \) |
| 7 | \( 1 + 0.0673T + 7T^{2} \) |
| 11 | \( 1 - 3.92T + 11T^{2} \) |
| 13 | \( 1 + 3.28T + 13T^{2} \) |
| 17 | \( 1 - 6.87T + 17T^{2} \) |
| 23 | \( 1 + 2.75iT - 23T^{2} \) |
| 29 | \( 1 - 6.62T + 29T^{2} \) |
| 31 | \( 1 + 8.36iT - 31T^{2} \) |
| 37 | \( 1 - 6.84T + 37T^{2} \) |
| 41 | \( 1 - 7.14iT - 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 - 6.02T + 47T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 9.76iT - 61T^{2} \) |
| 67 | \( 1 - 2.57iT - 67T^{2} \) |
| 71 | \( 1 - 3.31iT - 71T^{2} \) |
| 73 | \( 1 - 2.82iT - 73T^{2} \) |
| 79 | \( 1 + 4.59iT - 79T^{2} \) |
| 83 | \( 1 + 14.3iT - 83T^{2} \) |
| 89 | \( 1 + 7.42T + 89T^{2} \) |
| 97 | \( 1 + 8.36T + 97T^{2} \) |
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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
−8.019937335664847503411330770647, −7.38978002607830889491125878736, −6.71039344696783309692995324137, −6.26529443996750627885299611138, −5.42534183890081719658790294311, −4.41067368783099931571696911179, −3.14834756052186042793478386906, −2.50469304054540261282073904300, −1.41724280394250635653675481724, −0.67324706611363647570195611923,
1.13338787687082799949251304150, 2.74948133381808596282024742360, 3.49912222052056199615731520540, 4.10398882731016662809021643227, 5.06492475771283439121758282168, 5.33072448693299714550213933288, 6.40617213145161922299772875121, 7.24233948603280109088603118151, 8.341438608358589901417414062850, 8.795695804077535009470522769008
