L(s) = 1 | + i·3-s + 4·7-s + 2·9-s − 3i·11-s − 17-s + 7i·19-s + 4i·21-s + 4·23-s + 5i·27-s + 8i·29-s − 4·31-s + 3·33-s − 4i·37-s + 3·41-s − 8i·43-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 1.51·7-s + 0.666·9-s − 0.904i·11-s − 0.242·17-s + 1.60i·19-s + 0.872i·21-s + 0.834·23-s + 0.962i·27-s + 1.48i·29-s − 0.718·31-s + 0.522·33-s − 0.657i·37-s + 0.468·41-s − 1.21i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.480699497\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.480699497\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - iT - 3T^{2} \) |
| 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 - 7iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 8iT - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + 8iT - 59T^{2} \) |
| 61 | \( 1 + 4iT - 61T^{2} \) |
| 67 | \( 1 + 9iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + iT - 83T^{2} \) |
| 89 | \( 1 - 13T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−8.810374616972983077032101280482, −8.003468649372751867446624890246, −7.46421049216078911736682911635, −6.50052975836475833294090075358, −5.39860673005815620996313077596, −5.05068295593872540120114682017, −4.03242271446377246385216902292, −3.44857158452734810695733376748, −2.01304504756188878201251971742, −1.16475389322628628428977672982,
0.917373289275167210091064991340, 1.87734397217227271783701175354, 2.59838429583990112582170291791, 4.16056925622329383308980523986, 4.67974255918801202246969964115, 5.36071624779311946586659755746, 6.60279995630154838980504053932, 7.10411761831821489374647363360, 7.79641197254777268873573496253, 8.368244782725971002998394110140