| L(s) = 1 | + 0.841·3-s − 3.29·7-s − 2.29·9-s − 11-s − 13-s + 1.15·17-s − 1.31·19-s − 2.76·21-s + 3.84·23-s − 4.45·27-s + 6.61·29-s − 7.58·31-s − 0.841·33-s − 2.13·37-s − 0.841·39-s + 6.13·41-s + 8.81·43-s + 2.76·47-s + 3.84·49-s + 0.974·51-s + 10.1·53-s − 1.10·57-s − 2.76·59-s − 3.61·61-s + 7.54·63-s + 2.90·67-s + 3.23·69-s + ⋯ |
| L(s) = 1 | + 0.485·3-s − 1.24·7-s − 0.764·9-s − 0.301·11-s − 0.277·13-s + 0.281·17-s − 0.302·19-s − 0.604·21-s + 0.800·23-s − 0.856·27-s + 1.22·29-s − 1.36·31-s − 0.146·33-s − 0.350·37-s − 0.134·39-s + 0.957·41-s + 1.34·43-s + 0.403·47-s + 0.548·49-s + 0.136·51-s + 1.40·53-s − 0.146·57-s − 0.360·59-s − 0.462·61-s + 0.951·63-s + 0.354·67-s + 0.388·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.426088215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.426088215\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 0.841T + 3T^{2} \) |
| 7 | \( 1 + 3.29T + 7T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 1.15T + 17T^{2} \) |
| 19 | \( 1 + 1.31T + 19T^{2} \) |
| 23 | \( 1 - 3.84T + 23T^{2} \) |
| 29 | \( 1 - 6.61T + 29T^{2} \) |
| 31 | \( 1 + 7.58T + 31T^{2} \) |
| 37 | \( 1 + 2.13T + 37T^{2} \) |
| 41 | \( 1 - 6.13T + 41T^{2} \) |
| 43 | \( 1 - 8.81T + 43T^{2} \) |
| 47 | \( 1 - 2.76T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 2.76T + 59T^{2} \) |
| 61 | \( 1 + 3.61T + 61T^{2} \) |
| 67 | \( 1 - 2.90T + 67T^{2} \) |
| 71 | \( 1 + 0.866T + 71T^{2} \) |
| 73 | \( 1 + 5.87T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 8.92T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−8.499796190904944813884249249919, −7.57029346361662135394477736186, −7.01934943310234496166760715482, −6.08790685641803955600470608793, −5.59166176767550346852028953933, −4.55568974139826786448652295227, −3.55505702371458829722392108053, −2.95516537506063791040877134856, −2.23124593233089065195915240682, −0.62262407057399335576134861925,
0.62262407057399335576134861925, 2.23124593233089065195915240682, 2.95516537506063791040877134856, 3.55505702371458829722392108053, 4.55568974139826786448652295227, 5.59166176767550346852028953933, 6.08790685641803955600470608793, 7.01934943310234496166760715482, 7.57029346361662135394477736186, 8.499796190904944813884249249919
