| L(s) = 1 | − i·3-s + 2.23i·5-s − 2i·7-s − 9-s + 4.47·11-s + 4.47i·13-s + 2.23·15-s − 4.47i·17-s − 2·21-s + 4i·23-s − 5.00·25-s + i·27-s + 4·29-s + 8.94·31-s − 4.47i·33-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.999i·5-s − 0.755i·7-s − 0.333·9-s + 1.34·11-s + 1.24i·13-s + 0.577·15-s − 1.08i·17-s − 0.436·21-s + 0.834i·23-s − 1.00·25-s + 0.192i·27-s + 0.742·29-s + 1.60·31-s − 0.778i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.69574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69574\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.23iT \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 - 4.47iT - 13T^{2} \) |
| 17 | \( 1 + 4.47iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 8.94T + 31T^{2} \) |
| 37 | \( 1 - 4.47iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 4.47iT - 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 - 8.94iT - 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 17.8iT - 97T^{2} \) |
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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
−9.922350847022295434081800023545, −9.346456093007206887560157896780, −8.241080106975959788773053212237, −7.15421685859230362754743043280, −6.84053655056337720632191596926, −6.07660302350380684017213157500, −4.56853820826066625348322865034, −3.67601231974516185386114712585, −2.51930688287601154812676471111, −1.17669430306450095916000299573,
1.03314823935822114468338304043, 2.60784182038271971466460046772, 3.92560775896570845693483370407, 4.64976223466440426165275122913, 5.74633582915860319837464410985, 6.26442849528267361675624208511, 7.78948704067713622384607524440, 8.627612547157262560287324819809, 9.020219957143010272890835810334, 9.975594496554668053497489967154
