L(s) = 1 | + 1.73·3-s − 5.27·11-s + 2.62·13-s + 0.418·17-s − 3.27·19-s + 7.82·23-s − 5.19·27-s + 4.27·29-s − 3.27·31-s − 9.13·33-s + 9.97·37-s + 4.54·39-s + 3.72·41-s + 2.15·43-s + 6.50·47-s + 0.725·51-s + 5.67·53-s − 5.67·57-s + 3.27·59-s + 13.5·61-s − 3.52·67-s + 13.5·69-s − 4.54·71-s + 6.50·73-s + 7.27·79-s − 9·81-s − 7.40·83-s + ⋯ |
L(s) = 1 | + 1.00·3-s − 1.59·11-s + 0.728·13-s + 0.101·17-s − 0.751·19-s + 1.63·23-s − 1.00·27-s + 0.793·29-s − 0.588·31-s − 1.59·33-s + 1.63·37-s + 0.728·39-s + 0.581·41-s + 0.327·43-s + 0.949·47-s + 0.101·51-s + 0.779·53-s − 0.751·57-s + 0.426·59-s + 1.73·61-s − 0.430·67-s + 1.63·69-s − 0.539·71-s + 0.761·73-s + 0.818·79-s − 81-s − 0.812·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.500562249\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.500562249\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 11 | \( 1 + 5.27T + 11T^{2} \) |
| 13 | \( 1 - 2.62T + 13T^{2} \) |
| 17 | \( 1 - 0.418T + 17T^{2} \) |
| 19 | \( 1 + 3.27T + 19T^{2} \) |
| 23 | \( 1 - 7.82T + 23T^{2} \) |
| 29 | \( 1 - 4.27T + 29T^{2} \) |
| 31 | \( 1 + 3.27T + 31T^{2} \) |
| 37 | \( 1 - 9.97T + 37T^{2} \) |
| 41 | \( 1 - 3.72T + 41T^{2} \) |
| 43 | \( 1 - 2.15T + 43T^{2} \) |
| 47 | \( 1 - 6.50T + 47T^{2} \) |
| 53 | \( 1 - 5.67T + 53T^{2} \) |
| 59 | \( 1 - 3.27T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 + 3.52T + 67T^{2} \) |
| 71 | \( 1 + 4.54T + 71T^{2} \) |
| 73 | \( 1 - 6.50T + 73T^{2} \) |
| 79 | \( 1 - 7.27T + 79T^{2} \) |
| 83 | \( 1 + 7.40T + 83T^{2} \) |
| 89 | \( 1 - 7T + 89T^{2} \) |
| 97 | \( 1 + 6.92T + 97T^{2} \) |
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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
−8.326121027905072772811713769516, −7.70095685752843726620659018585, −7.03260177340731552516444411941, −6.01386146689088099326891195109, −5.36470254792710437397608647081, −4.47715449527884593327618765704, −3.57392938751520457393356303725, −2.73959997294310563654271200821, −2.28591405637713447536171358815, −0.812578734995134305732650201349,
0.812578734995134305732650201349, 2.28591405637713447536171358815, 2.73959997294310563654271200821, 3.57392938751520457393356303725, 4.47715449527884593327618765704, 5.36470254792710437397608647081, 6.01386146689088099326891195109, 7.03260177340731552516444411941, 7.70095685752843726620659018585, 8.326121027905072772811713769516