| L(s) = 1 | − i·3-s + (−1.89 − 1.18i)5-s − 4.20i·7-s − 9-s − 11-s − 2.90i·13-s + (−1.18 + 1.89i)15-s − 2.41i·17-s − 6.16·19-s − 4.20·21-s + 8.05i·23-s + (2.19 + 4.49i)25-s + i·27-s + 5.68·29-s + 1.52·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.848 − 0.529i)5-s − 1.58i·7-s − 0.333·9-s − 0.301·11-s − 0.804i·13-s + (−0.305 + 0.489i)15-s − 0.585i·17-s − 1.41·19-s − 0.917·21-s + 1.67i·23-s + (0.438 + 0.898i)25-s + 0.192i·27-s + 1.05·29-s + 0.273·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5965183159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5965183159\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.89 + 1.18i)T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 4.20iT - 7T^{2} \) |
| 13 | \( 1 + 2.90iT - 13T^{2} \) |
| 17 | \( 1 + 2.41iT - 17T^{2} \) |
| 19 | \( 1 + 6.16T + 19T^{2} \) |
| 23 | \( 1 - 8.05iT - 23T^{2} \) |
| 29 | \( 1 - 5.68T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 - 0.577iT - 37T^{2} \) |
| 41 | \( 1 - 3.30T + 41T^{2} \) |
| 43 | \( 1 + 10.6iT - 43T^{2} \) |
| 47 | \( 1 + 0.355iT - 47T^{2} \) |
| 53 | \( 1 - 9.95iT - 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 8.90T + 61T^{2} \) |
| 67 | \( 1 - 5.40iT - 67T^{2} \) |
| 71 | \( 1 - 6.61T + 71T^{2} \) |
| 73 | \( 1 - 6.21iT - 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 10.7iT - 83T^{2} \) |
| 89 | \( 1 + 7.52T + 89T^{2} \) |
| 97 | \( 1 - 3.33iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.983816211202246066379738589154, −8.135429522343422851480260021407, −7.50432389549228349771855314591, −7.02326788442823698069241389314, −5.84864328016698904943527096700, −4.74590772350914974716326111534, −3.99227897769908756969761962892, −3.00166266735070065746905819549, −1.32260103586234665145656891242, −0.25756743806928078009709982620,
2.24427361220912339731730182142, 2.98826002543314049682404789603, 4.25283259754301588656620124051, 4.83027951515077042349212764689, 6.21302965367055574824252607822, 6.50445271082852464531645814025, 7.992723719607109670433338616025, 8.509131147322587580971286097414, 9.144457357324155084261144341266, 10.20039873826528918230133508866
