| L(s) = 1 | + 2.23i·5-s + 2i·7-s − 4.47·11-s − 4.47i·13-s + 4.47i·17-s + 4i·23-s − 5.00·25-s + 4·29-s − 8.94·31-s − 4.47·35-s − 4.47i·37-s − 10·41-s + 4i·43-s − 8i·47-s + 3·49-s + ⋯ |
| L(s) = 1 | + 0.999i·5-s + 0.755i·7-s − 1.34·11-s − 1.24i·13-s + 1.08i·17-s + 0.834i·23-s − 1.00·25-s + 0.742·29-s − 1.60·31-s − 0.755·35-s − 0.735i·37-s − 1.56·41-s + 0.609i·43-s − 1.16i·47-s + 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5288574375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5288574375\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 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
−10.28050954446112390869298658382, −9.117742768646456005761218017698, −8.135649142783730366330901440000, −7.67220441691546342250272553804, −6.67009507343241820872957833098, −5.66412235980262410291366344682, −5.27728529715644030126361410230, −3.69317192137638188340017732954, −2.93279612884887630196428836064, −1.99594864537663159439266731873,
0.19907886645495187645100147373, 1.63197097223259726290013129297, 2.90055613177268380447918844684, 4.24156084630314207135651686601, 4.80623187152615436896213555662, 5.65262436587696134203888218539, 6.85469065811911936852994294426, 7.50031055511608455646562200067, 8.396966278667918173498469166484, 9.082150021756316601308729261382
