| L(s) = 1 | − i·3-s + (0.432 − 2.19i)5-s − 9-s − 2.86·11-s − i·13-s + (−2.19 − 0.432i)15-s − 5.52i·17-s − 3.52·19-s + 7.52i·23-s + (−4.62 − 1.89i)25-s + i·27-s − 6.77·29-s + 5.72·31-s + 2.86i·33-s + 3.72i·37-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (0.193 − 0.981i)5-s − 0.333·9-s − 0.863·11-s − 0.277i·13-s + (−0.566 − 0.111i)15-s − 1.33i·17-s − 0.808·19-s + 1.56i·23-s + (−0.925 − 0.379i)25-s + 0.192i·27-s − 1.25·29-s + 1.02·31-s + 0.498i·33-s + 0.613i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7038444852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7038444852\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-0.432 + 2.19i)T \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 2.86T + 11T^{2} \) |
| 17 | \( 1 + 5.52iT - 17T^{2} \) |
| 19 | \( 1 + 3.52T + 19T^{2} \) |
| 23 | \( 1 - 7.52iT - 23T^{2} \) |
| 29 | \( 1 + 6.77T + 29T^{2} \) |
| 31 | \( 1 - 5.72T + 31T^{2} \) |
| 37 | \( 1 - 3.72iT - 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 5.52iT - 43T^{2} \) |
| 47 | \( 1 + 8.65iT - 47T^{2} \) |
| 53 | \( 1 + 6.77iT - 53T^{2} \) |
| 59 | \( 1 + 0.593T + 59T^{2} \) |
| 61 | \( 1 + 5.25T + 61T^{2} \) |
| 67 | \( 1 - 10.5iT - 67T^{2} \) |
| 71 | \( 1 + 2.38T + 71T^{2} \) |
| 73 | \( 1 - 5.45iT - 73T^{2} \) |
| 79 | \( 1 + 2.47T + 79T^{2} \) |
| 83 | \( 1 - 8.11iT - 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−8.923901756996164232770234598811, −8.201897937956572162333721948986, −7.50358735574620922758862165661, −6.67206910552901408641580288668, −5.41203491320019935657354033754, −5.22313858781190639101729956970, −3.91008759763318696290211277627, −2.68119735105710948267816707283, −1.61383289075840042996961143375, −0.25651970391370069379619951743,
2.02484172789241170997944818127, 2.92970900312235734574531812403, 3.94731023187876966275697102425, 4.78269923429936598028536343139, 5.98888509418124426703172247845, 6.41076354955591419663032683115, 7.53319639947653276167057760273, 8.278135692538432201663442567797, 9.105170335525612336289032606322, 10.07603361051203060104647849857
