| L(s) = 1 | − 7-s + 5.19·11-s − 2·13-s + 2·19-s − 5.19·23-s + 5.19·29-s + 2·31-s + 37-s − 5·43-s + 10.3·47-s + 49-s − 10.3·59-s + 8·61-s + 7·67-s − 5.19·71-s − 8·73-s − 5.19·77-s + 5·79-s + 10.3·83-s + 10.3·89-s + 2·91-s + 10·97-s − 2·103-s − 10.3·107-s − 109-s + 5.19·113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.56·11-s − 0.554·13-s + 0.458·19-s − 1.08·23-s + 0.964·29-s + 0.359·31-s + 0.164·37-s − 0.762·43-s + 1.51·47-s + 0.142·49-s − 1.35·59-s + 1.02·61-s + 0.855·67-s − 0.616·71-s − 0.936·73-s − 0.592·77-s + 0.562·79-s + 1.14·83-s + 1.10·89-s + 0.209·91-s + 1.01·97-s − 0.197·103-s − 1.00·107-s − 0.0957·109-s + 0.488·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.995904506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.995904506\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 5.19T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 5.19T + 23T^{2} \) |
| 29 | \( 1 - 5.19T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 7T + 67T^{2} \) |
| 71 | \( 1 + 5.19T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.009453558555650146854530908810, −7.29052118447827194730304903333, −6.53653295086721472746579570588, −6.11635716448353117977695011511, −5.15933903721764150552906732447, −4.31082379899903763448916238423, −3.70105857299602315751269251191, −2.80070195812088469099053808016, −1.80252283893987559868457194371, −0.75048225872808911157737227977,
0.75048225872808911157737227977, 1.80252283893987559868457194371, 2.80070195812088469099053808016, 3.70105857299602315751269251191, 4.31082379899903763448916238423, 5.15933903721764150552906732447, 6.11635716448353117977695011511, 6.53653295086721472746579570588, 7.29052118447827194730304903333, 8.009453558555650146854530908810
