L(s) = 1 | + (0.0475 + 0.998i)3-s + (−0.654 − 0.755i)4-s + (0.235 − 0.971i)7-s + (−0.995 + 0.0950i)9-s + (0.723 − 0.690i)12-s + (0.601 − 0.573i)13-s + (−0.142 + 0.989i)16-s + (−0.738 − 1.61i)19-s + (0.981 + 0.189i)21-s + (0.723 − 0.690i)25-s + (−0.142 − 0.989i)27-s + (−0.888 + 0.458i)28-s + (0.273 − 0.0801i)31-s + (0.723 + 0.690i)36-s + 0.471·37-s + ⋯ |
L(s) = 1 | + (0.0475 + 0.998i)3-s + (−0.654 − 0.755i)4-s + (0.235 − 0.971i)7-s + (−0.995 + 0.0950i)9-s + (0.723 − 0.690i)12-s + (0.601 − 0.573i)13-s + (−0.142 + 0.989i)16-s + (−0.738 − 1.61i)19-s + (0.981 + 0.189i)21-s + (0.723 − 0.690i)25-s + (−0.142 − 0.989i)27-s + (−0.888 + 0.458i)28-s + (0.273 − 0.0801i)31-s + (0.723 + 0.690i)36-s + 0.471·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8799829574\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8799829574\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.0475 - 0.998i)T \) |
| 7 | \( 1 + (-0.235 + 0.971i)T \) |
| 67 | \( 1 + (-0.235 - 0.971i)T \) |
good | 2 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 5 | \( 1 + (-0.723 + 0.690i)T^{2} \) |
| 11 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 13 | \( 1 + (-0.601 + 0.573i)T + (0.0475 - 0.998i)T^{2} \) |
| 17 | \( 1 + (-0.928 - 0.371i)T^{2} \) |
| 19 | \( 1 + (0.738 + 1.61i)T + (-0.654 + 0.755i)T^{2} \) |
| 23 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \) |
| 37 | \( 1 - 0.471T + T^{2} \) |
| 41 | \( 1 + (-0.928 - 0.371i)T^{2} \) |
| 43 | \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \) |
| 47 | \( 1 + (0.995 - 0.0950i)T^{2} \) |
| 53 | \( 1 + (-0.928 + 0.371i)T^{2} \) |
| 59 | \( 1 + (0.888 + 0.458i)T^{2} \) |
| 61 | \( 1 + (-0.252 - 1.75i)T + (-0.959 + 0.281i)T^{2} \) |
| 71 | \( 1 + (0.786 + 0.618i)T^{2} \) |
| 73 | \( 1 + (1.45 - 1.14i)T + (0.235 - 0.971i)T^{2} \) |
| 79 | \( 1 + (-0.341 + 0.325i)T + (0.0475 - 0.998i)T^{2} \) |
| 83 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 89 | \( 1 + (0.995 + 0.0950i)T^{2} \) |
| 97 | \( 1 + (0.0475 - 0.0824i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.755781322050503076575268351953, −8.841784835373826744705345538316, −8.451227052905097668451281167257, −7.19005554283685262820096334754, −6.18090641834153463018637601963, −5.31047383581860312223219281495, −4.47533463981757084797213883066, −3.97765434630225987458361541324, −2.65359040985140493965634857432, −0.789305932020606694103965802392,
1.58705271891404224261107386708, 2.73189526782186096002377238690, 3.72138331792197853799744341968, 4.85738095195580844886064367039, 5.88534046677350688486029879031, 6.54185678394448796375980573281, 7.72205385520305486193607404762, 8.157777420602681767184277555745, 8.903402598615596080364379394684, 9.467269842717169346454813421908