L(s) = 1 | − 0.888·2-s − 1.39·3-s − 1.21·4-s + 1.07·5-s + 1.23·6-s − 3.99·7-s + 2.85·8-s − 1.06·9-s − 0.953·10-s − 2.51·11-s + 1.68·12-s − 4.30·13-s + 3.55·14-s − 1.49·15-s − 0.111·16-s − 4.78·17-s + 0.946·18-s − 0.979·19-s − 1.29·20-s + 5.56·21-s + 2.23·22-s − 2.20·23-s − 3.96·24-s − 3.84·25-s + 3.82·26-s + 5.65·27-s + 4.84·28-s + ⋯ |
L(s) = 1 | − 0.628·2-s − 0.803·3-s − 0.605·4-s + 0.479·5-s + 0.504·6-s − 1.51·7-s + 1.00·8-s − 0.355·9-s − 0.301·10-s − 0.757·11-s + 0.486·12-s − 1.19·13-s + 0.949·14-s − 0.385·15-s − 0.0279·16-s − 1.15·17-s + 0.223·18-s − 0.224·19-s − 0.290·20-s + 1.21·21-s + 0.476·22-s − 0.459·23-s − 0.809·24-s − 0.769·25-s + 0.749·26-s + 1.08·27-s + 0.915·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1196607908\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1196607908\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 41 | \( 1 \) |
good | 2 | \( 1 + 0.888T + 2T^{2} \) |
| 3 | \( 1 + 1.39T + 3T^{2} \) |
| 5 | \( 1 - 1.07T + 5T^{2} \) |
| 7 | \( 1 + 3.99T + 7T^{2} \) |
| 11 | \( 1 + 2.51T + 11T^{2} \) |
| 13 | \( 1 + 4.30T + 13T^{2} \) |
| 17 | \( 1 + 4.78T + 17T^{2} \) |
| 19 | \( 1 + 0.979T + 19T^{2} \) |
| 23 | \( 1 + 2.20T + 23T^{2} \) |
| 29 | \( 1 - 2.05T + 29T^{2} \) |
| 31 | \( 1 - 6.55T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 43 | \( 1 + 2.01T + 43T^{2} \) |
| 47 | \( 1 + 9.95T + 47T^{2} \) |
| 53 | \( 1 + 7.76T + 53T^{2} \) |
| 59 | \( 1 + 9.60T + 59T^{2} \) |
| 61 | \( 1 - 0.448T + 61T^{2} \) |
| 67 | \( 1 - 3.50T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 - 8.75T + 73T^{2} \) |
| 79 | \( 1 - 2.73T + 79T^{2} \) |
| 83 | \( 1 + 2.79T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 3.76T + 97T^{2} \) |
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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
−9.548211317604552143383230065557, −8.698961122385390196360067358189, −7.891293191982947814621548832001, −6.78392700084769790251178389536, −6.26003169150286564836484066606, −5.25083847054244554265428844896, −4.63057277998613156674641685108, −3.28896646716567776938946546160, −2.17699906405078470583377651237, −0.25584817450458880102199553537,
0.25584817450458880102199553537, 2.17699906405078470583377651237, 3.28896646716567776938946546160, 4.63057277998613156674641685108, 5.25083847054244554265428844896, 6.26003169150286564836484066606, 6.78392700084769790251178389536, 7.891293191982947814621548832001, 8.698961122385390196360067358189, 9.548211317604552143383230065557