| L(s) = 1 | − 3.16i·3-s + 2.23i·5-s + (−2.12 − 1.58i)7-s − 7.00·9-s + 7.07·15-s + (−5.00 + 6.70i)21-s − 1.41·23-s − 5.00·25-s + 12.6i·27-s − 6·29-s + (3.53 − 4.74i)35-s + 4.47i·41-s + 12.7·43-s − 15.6i·45-s + 9.48i·47-s + ⋯ |
| L(s) = 1 | − 1.82i·3-s + 0.999i·5-s + (−0.801 − 0.597i)7-s − 2.33·9-s + 1.82·15-s + (−1.09 + 1.46i)21-s − 0.294·23-s − 1.00·25-s + 2.43i·27-s − 1.11·29-s + (0.597 − 0.801i)35-s + 0.698i·41-s + 1.94·43-s − 2.33i·45-s + 1.38i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 - 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.597 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5170329201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5170329201\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 + (2.12 + 1.58i)T \) |
| good | 3 | \( 1 + 3.16iT - 3T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 4.47iT - 41T^{2} \) |
| 43 | \( 1 - 12.7T + 43T^{2} \) |
| 47 | \( 1 - 9.48iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 13.4iT - 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 9.48iT - 83T^{2} \) |
| 89 | \( 1 - 17.8iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
| 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
−9.068057338873922947856273802069, −7.984538948680692690551687800366, −7.42623752358257064788204371798, −6.96631795292784119065608923740, −6.18626744637382571576502939087, −5.74822505810721861857172927406, −4.06386846384179583269468146414, −3.03154078405596798249375947804, −2.35552498958665531634317790944, −1.15553645329425356869077312689,
0.18816949242081315049372663499, 2.25787638220547326247290733252, 3.43937411933734357360671393601, 4.02180998818222127083470969729, 4.91513420760546730349229050473, 5.55690075865132424130605400543, 6.16199721231586137252186261485, 7.57874540771403946438293271777, 8.636545964444275892544645079451, 9.029535650293753867293623996456
