| L(s) = 1 | + 2·7-s + 4i·11-s + 6i·13-s − 6·17-s − 4·23-s + 5·25-s − 4i·29-s − 10·31-s − 2i·37-s − 2·41-s + 8i·43-s − 12·47-s − 3·49-s − 12i·53-s + 4i·59-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 1.20i·11-s + 1.66i·13-s − 1.45·17-s − 0.834·23-s + 25-s − 0.742i·29-s − 1.79·31-s − 0.328i·37-s − 0.312·41-s + 1.21i·43-s − 1.75·47-s − 0.428·49-s − 1.64i·53-s + 0.520i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.033206410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.033206410\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 2iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−9.347326338096017142438418932643, −8.552507796422888236654309077914, −7.76083197203837316097170918019, −6.84879269632144457025565923293, −6.46911112005204211609772978174, −5.08068589332762480492005333298, −4.55539523296991156612474715791, −3.81354758989909082739341563029, −2.19758761057679051211430770345, −1.75287322298532242533148896277,
0.32950532247544385902251709809, 1.70345657787929674360977736431, 2.91502579540998992668842195212, 3.70820004112830324720460019691, 4.88059365848098470921519903672, 5.47744472932571929058752740041, 6.32367353764861735640204142475, 7.23822972083437491706903627209, 8.154492849090770594003693589834, 8.534186416259772447699450274897
