L(s) = 1 | + 2-s − 0.339·3-s − 4-s − 2.88·5-s − 0.339·6-s − 3.54·7-s − 3·8-s − 2.88·9-s − 2.88·10-s + 11-s + 0.339·12-s − 3.54·14-s + 0.980·15-s − 16-s − 5.22·17-s − 2.88·18-s + 6.22·19-s + 2.88·20-s + 1.20·21-s + 22-s + 0.679·23-s + 1.01·24-s + 3.32·25-s + 2·27-s + 3.54·28-s − 8.42·29-s + 0.980·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.196·3-s − 0.5·4-s − 1.28·5-s − 0.138·6-s − 1.33·7-s − 1.06·8-s − 0.961·9-s − 0.912·10-s + 0.301·11-s + 0.0981·12-s − 0.947·14-s + 0.253·15-s − 0.250·16-s − 1.26·17-s − 0.679·18-s + 1.42·19-s + 0.644·20-s + 0.262·21-s + 0.213·22-s + 0.141·23-s + 0.208·24-s + 0.664·25-s + 0.384·27-s + 0.669·28-s − 1.56·29-s + 0.178·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5404455784\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5404455784\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - T + 2T^{2} \) |
| 3 | \( 1 + 0.339T + 3T^{2} \) |
| 5 | \( 1 + 2.88T + 5T^{2} \) |
| 7 | \( 1 + 3.54T + 7T^{2} \) |
| 17 | \( 1 + 5.22T + 17T^{2} \) |
| 19 | \( 1 - 6.22T + 19T^{2} \) |
| 23 | \( 1 - 0.679T + 23T^{2} \) |
| 29 | \( 1 + 8.42T + 29T^{2} \) |
| 31 | \( 1 - 8.40T + 31T^{2} \) |
| 37 | \( 1 - 3.52T + 37T^{2} \) |
| 41 | \( 1 + 5.74T + 41T^{2} \) |
| 43 | \( 1 + 1.32T + 43T^{2} \) |
| 47 | \( 1 - 6.10T + 47T^{2} \) |
| 53 | \( 1 + 6.08T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 4.01T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 79 | \( 1 - 0.904T + 79T^{2} \) |
| 83 | \( 1 - 4.90T + 83T^{2} \) |
| 89 | \( 1 + 5.65T + 89T^{2} \) |
| 97 | \( 1 - 2.79T + 97T^{2} \) |
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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
−9.211955313167439162077185048399, −8.528064760868993836830625101141, −7.63695046355354644934752246336, −6.66324414536037001479459077793, −6.00420939013043536816905355584, −5.08378612631823373345260527427, −4.18464834660147873618149790435, −3.44362640868295960425022357228, −2.85377984765603891606613705573, −0.42758339549553748542860686006,
0.42758339549553748542860686006, 2.85377984765603891606613705573, 3.44362640868295960425022357228, 4.18464834660147873618149790435, 5.08378612631823373345260527427, 6.00420939013043536816905355584, 6.66324414536037001479459077793, 7.63695046355354644934752246336, 8.528064760868993836830625101141, 9.211955313167439162077185048399