| L(s) = 1 | + 0.954i·5-s + 4.78i·7-s + 2.64i·11-s + (−1.58 − 3.23i)13-s − 4.47·17-s + 2.30i·19-s − 8.38·23-s + 4.08·25-s + 3.90·29-s + 2.87i·31-s − 4.56·35-s + 8.38i·37-s − 4.95i·41-s + 7.65·43-s − 10.2i·47-s + ⋯ |
| L(s) = 1 | + 0.426i·5-s + 1.80i·7-s + 0.797i·11-s + (−0.440 − 0.897i)13-s − 1.08·17-s + 0.529i·19-s − 1.74·23-s + 0.817·25-s + 0.725·29-s + 0.515i·31-s − 0.771·35-s + 1.37i·37-s − 0.773i·41-s + 1.16·43-s − 1.50i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 + 0.440i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.897 + 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5746812669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5746812669\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.58 + 3.23i)T \) |
| good | 5 | \( 1 - 0.954iT - 5T^{2} \) |
| 7 | \( 1 - 4.78iT - 7T^{2} \) |
| 11 | \( 1 - 2.64iT - 11T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 2.30iT - 19T^{2} \) |
| 23 | \( 1 + 8.38T + 23T^{2} \) |
| 29 | \( 1 - 3.90T + 29T^{2} \) |
| 31 | \( 1 - 2.87iT - 31T^{2} \) |
| 37 | \( 1 - 8.38iT - 37T^{2} \) |
| 41 | \( 1 + 4.95iT - 41T^{2} \) |
| 43 | \( 1 - 7.65T + 43T^{2} \) |
| 47 | \( 1 + 10.2iT - 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 4.55iT - 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 - 0.399iT - 67T^{2} \) |
| 71 | \( 1 + 3.44iT - 71T^{2} \) |
| 73 | \( 1 + 12.2iT - 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 - 0.737iT - 83T^{2} \) |
| 89 | \( 1 - 11.8iT - 89T^{2} \) |
| 97 | \( 1 + 6.09iT - 97T^{2} \) |
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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
−8.876566345792505862808624816118, −8.257779856886462130125801214368, −7.56185457605921276134828636413, −6.52560187481002079230885848638, −6.06452364902640859157994313584, −5.18248685224331756181338990549, −4.54667754779750031647309537346, −3.28592145838318987321892784176, −2.50326991710910555534679048518, −1.85869075633145249736217002633,
0.16946619656698875537004111819, 1.18518292145611138449366846886, 2.37995749996569907353676688724, 3.57422509261004283285450455139, 4.41977184171070157351763478240, 4.65656606813623576895854164910, 6.07798934297874494372674817339, 6.58185557609460134393796654761, 7.44847626259831206751236468877, 7.947445517040960043385697111125
