L(s) = 1 | − 2.82·5-s − 6·11-s − 5.65·13-s − 1.41·17-s − 4.24·19-s − 4·23-s + 3.00·25-s + 6·29-s + 2.82·31-s + 2·37-s + 1.41·41-s + 10·43-s + 2.82·47-s + 2·53-s + 16.9·55-s − 1.41·59-s − 8.48·61-s + 16.0·65-s + 4·67-s + 12·71-s − 9.89·73-s − 4·79-s − 1.41·83-s + 4.00·85-s + 4.24·89-s + 12·95-s + 12.7·97-s + ⋯ |
L(s) = 1 | − 1.26·5-s − 1.80·11-s − 1.56·13-s − 0.342·17-s − 0.973·19-s − 0.834·23-s + 0.600·25-s + 1.11·29-s + 0.508·31-s + 0.328·37-s + 0.220·41-s + 1.52·43-s + 0.412·47-s + 0.274·53-s + 2.28·55-s − 0.184·59-s − 1.08·61-s + 1.98·65-s + 0.488·67-s + 1.42·71-s − 1.15·73-s − 0.450·79-s − 0.155·83-s + 0.433·85-s + 0.449·89-s + 1.23·95-s + 1.29·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5081538485\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5081538485\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 1.41T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + 8.48T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 9.89T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 - 4.24T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−8.282450084023836002208102668313, −7.82623837176705370339088250099, −7.37019172373713341674979438395, −6.42702747481272562963285536877, −5.40917985134451442849753675087, −4.62675643845550271138437381653, −4.11528239003647898410515933592, −2.83672231993995609620517675329, −2.33245980035607948818486635342, −0.38859476264435154329064331849,
0.38859476264435154329064331849, 2.33245980035607948818486635342, 2.83672231993995609620517675329, 4.11528239003647898410515933592, 4.62675643845550271138437381653, 5.40917985134451442849753675087, 6.42702747481272562963285536877, 7.37019172373713341674979438395, 7.82623837176705370339088250099, 8.282450084023836002208102668313