L(s) = 1 | + 2-s + 0.761·3-s + 4-s + 0.666·5-s + 0.761·6-s − 1.04·7-s + 8-s − 2.41·9-s + 0.666·10-s + 5.59·11-s + 0.761·12-s + 5.65·13-s − 1.04·14-s + 0.507·15-s + 16-s + 2.95·17-s − 2.41·18-s − 7.03·19-s + 0.666·20-s − 0.797·21-s + 5.59·22-s + 7.83·23-s + 0.761·24-s − 4.55·25-s + 5.65·26-s − 4.12·27-s − 1.04·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.439·3-s + 0.5·4-s + 0.297·5-s + 0.310·6-s − 0.395·7-s + 0.353·8-s − 0.806·9-s + 0.210·10-s + 1.68·11-s + 0.219·12-s + 1.56·13-s − 0.279·14-s + 0.131·15-s + 0.250·16-s + 0.717·17-s − 0.570·18-s − 1.61·19-s + 0.148·20-s − 0.174·21-s + 1.19·22-s + 1.63·23-s + 0.155·24-s − 0.911·25-s + 1.10·26-s − 0.794·27-s − 0.197·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.101924923\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.101924923\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
good | 3 | \( 1 - 0.761T + 3T^{2} \) |
| 5 | \( 1 - 0.666T + 5T^{2} \) |
| 7 | \( 1 + 1.04T + 7T^{2} \) |
| 11 | \( 1 - 5.59T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 - 2.95T + 17T^{2} \) |
| 19 | \( 1 + 7.03T + 19T^{2} \) |
| 23 | \( 1 - 7.83T + 23T^{2} \) |
| 29 | \( 1 + 6.27T + 29T^{2} \) |
| 31 | \( 1 - 5.53T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 2.70T + 41T^{2} \) |
| 43 | \( 1 + 9.38T + 43T^{2} \) |
| 47 | \( 1 - 4.88T + 47T^{2} \) |
| 53 | \( 1 - 3.98T + 53T^{2} \) |
| 59 | \( 1 - 1.28T + 59T^{2} \) |
| 61 | \( 1 + 2.84T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 + 3.54T + 73T^{2} \) |
| 79 | \( 1 - 8.00T + 79T^{2} \) |
| 83 | \( 1 + 0.521T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 - 2.41T + 97T^{2} \) |
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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
−8.541489069543443444746703394075, −7.73739808625497890112768747526, −6.60549791432461589967169961302, −6.24233654850864052531906869132, −5.68273663699535104234145156563, −4.47699436992936741546912279379, −3.72573794956768045653528805671, −3.21191934756296576628803361709, −2.11752118017256374851224261768, −1.09743226725543604043978902483,
1.09743226725543604043978902483, 2.11752118017256374851224261768, 3.21191934756296576628803361709, 3.72573794956768045653528805671, 4.47699436992936741546912279379, 5.68273663699535104234145156563, 6.24233654850864052531906869132, 6.60549791432461589967169961302, 7.73739808625497890112768747526, 8.541489069543443444746703394075