L(s) = 1 | + (−1.29 − 0.400i)3-s + (0.733 − 0.680i)5-s + (0.149 + 0.988i)7-s + (0.702 + 0.479i)9-s + (−1.22 + 0.590i)15-s + (0.202 − 1.34i)21-s + (0.215 − 0.548i)23-s + (0.0747 − 0.997i)25-s + (0.126 + 0.158i)27-s + (0.455 − 0.571i)29-s + (0.781 + 0.623i)35-s + (0.367 + 1.61i)41-s + (0.250 − 1.09i)43-s + (0.841 − 0.126i)45-s + (0.145 + 1.94i)47-s + ⋯ |
L(s) = 1 | + (−1.29 − 0.400i)3-s + (0.733 − 0.680i)5-s + (0.149 + 0.988i)7-s + (0.702 + 0.479i)9-s + (−1.22 + 0.590i)15-s + (0.202 − 1.34i)21-s + (0.215 − 0.548i)23-s + (0.0747 − 0.997i)25-s + (0.126 + 0.158i)27-s + (0.455 − 0.571i)29-s + (0.781 + 0.623i)35-s + (0.367 + 1.61i)41-s + (0.250 − 1.09i)43-s + (0.841 − 0.126i)45-s + (0.145 + 1.94i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9441785358\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9441785358\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.733 + 0.680i)T \) |
| 7 | \( 1 + (-0.149 - 0.988i)T \) |
good | 3 | \( 1 + (1.29 + 0.400i)T + (0.826 + 0.563i)T^{2} \) |
| 11 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 13 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 17 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.215 + 0.548i)T + (-0.733 - 0.680i)T^{2} \) |
| 29 | \( 1 + (-0.455 + 0.571i)T + (-0.222 - 0.974i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 41 | \( 1 + (-0.367 - 1.61i)T + (-0.900 + 0.433i)T^{2} \) |
| 43 | \( 1 + (-0.250 + 1.09i)T + (-0.900 - 0.433i)T^{2} \) |
| 47 | \( 1 + (-0.145 - 1.94i)T + (-0.988 + 0.149i)T^{2} \) |
| 53 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 59 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 61 | \( 1 + (-1.88 - 0.284i)T + (0.955 + 0.294i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 73 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.268 + 0.129i)T + (0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (-0.603 - 0.411i)T + (0.365 + 0.930i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.585988824533035873654298075532, −7.923350712217258755304148191122, −6.79399481111414878091499546458, −6.19734898346170237171040946442, −5.69490145352546708263952874758, −5.03709871028885658586151261163, −4.40566407700974273459588505800, −2.86698921553906637747377270708, −1.92688922137166168500361429676, −0.866500197665559404185357056248,
0.942371759165081753745261452808, 2.21108037965749714794135500965, 3.44077312603175030599951851343, 4.22201115917508334472930044761, 5.23090384641782329696336863750, 5.56632377137183185460220657353, 6.61828746375355697429063844119, 6.89238405435638392968830045592, 7.78348379994622968882589537949, 8.825202293650485360094904696133