L(s) = 1 | + 2-s + 3-s + 4-s + 0.692·5-s + 6-s − 0.356·7-s + 8-s + 9-s + 0.692·10-s − 2.93·11-s + 12-s − 0.356·14-s + 0.692·15-s + 16-s + 6.71·17-s + 18-s + 7.20·19-s + 0.692·20-s − 0.356·21-s − 2.93·22-s + 2.39·23-s + 24-s − 4.52·25-s + 27-s − 0.356·28-s + 7.82·29-s + 0.692·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.309·5-s + 0.408·6-s − 0.134·7-s + 0.353·8-s + 0.333·9-s + 0.218·10-s − 0.886·11-s + 0.288·12-s − 0.0953·14-s + 0.178·15-s + 0.250·16-s + 1.62·17-s + 0.235·18-s + 1.65·19-s + 0.154·20-s − 0.0778·21-s − 0.626·22-s + 0.499·23-s + 0.204·24-s − 0.904·25-s + 0.192·27-s − 0.0674·28-s + 1.45·29-s + 0.126·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.210294050\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.210294050\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 0.692T + 5T^{2} \) |
| 7 | \( 1 + 0.356T + 7T^{2} \) |
| 11 | \( 1 + 2.93T + 11T^{2} \) |
| 17 | \( 1 - 6.71T + 17T^{2} \) |
| 19 | \( 1 - 7.20T + 19T^{2} \) |
| 23 | \( 1 - 2.39T + 23T^{2} \) |
| 29 | \( 1 - 7.82T + 29T^{2} \) |
| 31 | \( 1 - 2.76T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 4.89T + 41T^{2} \) |
| 43 | \( 1 - 6.59T + 43T^{2} \) |
| 47 | \( 1 + 4.98T + 47T^{2} \) |
| 53 | \( 1 + 8.88T + 53T^{2} \) |
| 59 | \( 1 + 1.64T + 59T^{2} \) |
| 61 | \( 1 + 6.49T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 6.81T + 71T^{2} \) |
| 73 | \( 1 + 3.18T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 + 0.396T + 89T^{2} \) |
| 97 | \( 1 - 0.417T + 97T^{2} \) |
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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
−10.03087626637773667181130769081, −9.258956647684940162198927378293, −8.005866956444706997993098420887, −7.58572510315508435458140344308, −6.48446213074067317772895634041, −5.46765643659823242065605418549, −4.83993880642181214309290634499, −3.39677022724100594602251071794, −2.90383236945367784484430532902, −1.44507304698032299892028334665,
1.44507304698032299892028334665, 2.90383236945367784484430532902, 3.39677022724100594602251071794, 4.83993880642181214309290634499, 5.46765643659823242065605418549, 6.48446213074067317772895634041, 7.58572510315508435458140344308, 8.005866956444706997993098420887, 9.258956647684940162198927378293, 10.03087626637773667181130769081