| L(s) = 1 | − i·2-s − 4-s − 3.90·5-s + (1.49 − 2.18i)7-s + i·8-s + 3.90i·10-s − i·11-s − 6.17i·13-s + (−2.18 − 1.49i)14-s + 16-s − 0.462·17-s − 1.61i·19-s + 3.90·20-s − 22-s + 10.2·25-s − 6.17·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 1.74·5-s + (0.565 − 0.824i)7-s + 0.353i·8-s + 1.23i·10-s − 0.301i·11-s − 1.71i·13-s + (−0.583 − 0.399i)14-s + 0.250·16-s − 0.112·17-s − 0.370i·19-s + 0.872·20-s − 0.213·22-s + 2.04·25-s − 1.21·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.347 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1913273309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1913273309\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.49 + 2.18i)T \) |
| 11 | \( 1 + iT \) |
| good | 5 | \( 1 + 3.90T + 5T^{2} \) |
| 13 | \( 1 + 6.17iT - 13T^{2} \) |
| 17 | \( 1 + 0.462T + 17T^{2} \) |
| 19 | \( 1 + 1.61iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 7.80iT - 29T^{2} \) |
| 31 | \( 1 - 5.98iT - 31T^{2} \) |
| 37 | \( 1 + 9.05T + 37T^{2} \) |
| 41 | \( 1 + 0.191T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + 5.05T + 47T^{2} \) |
| 53 | \( 1 - 6.04iT - 53T^{2} \) |
| 59 | \( 1 + 4.54T + 59T^{2} \) |
| 61 | \( 1 - 2.55iT - 61T^{2} \) |
| 67 | \( 1 + 5.80T + 67T^{2} \) |
| 71 | \( 1 + 8.48iT - 71T^{2} \) |
| 73 | \( 1 - 10.8iT - 73T^{2} \) |
| 79 | \( 1 - 16.2T + 79T^{2} \) |
| 83 | \( 1 + 9.86T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 11.0iT - 97T^{2} \) |
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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
−8.757176643959136582537123128958, −8.288333007457378550837077387284, −7.56837465523684057750711201377, −6.86717522528454225245333311777, −5.23447159831019478903719238188, −4.65336494174402402819095501586, −3.51186584384915743186380866460, −3.18193248773339293311275740915, −1.23478675821366208380600099169, −0.087808119439193485854068825502,
1.93657718648323871182440296832, 3.51757931523650794019335207339, 4.35748476031605493203187442497, 4.93137097731542300632769700998, 6.19617320031198080135258750314, 6.99548350910014455076821091330, 7.76909702409969388758625588075, 8.373883325553827387762376404129, 9.006381551807765807713787742245, 9.922857105678525748465826046727
