| L(s) = 1 | − 2-s − 3-s + 4-s − 4.06·5-s + 6-s − 8-s − 2·9-s + 4.06·10-s − 4.67·11-s − 12-s − 1.86·13-s + 4.06·15-s + 16-s + 2.54·17-s + 2·18-s + 6.45·19-s − 4.06·20-s + 4.67·22-s − 6.32·23-s + 24-s + 11.5·25-s + 1.86·26-s + 5·27-s + 0.932·29-s − 4.06·30-s + 8.38·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.81·5-s + 0.408·6-s − 0.353·8-s − 0.666·9-s + 1.28·10-s − 1.40·11-s − 0.288·12-s − 0.517·13-s + 1.04·15-s + 0.250·16-s + 0.617·17-s + 0.471·18-s + 1.48·19-s − 0.909·20-s + 0.995·22-s − 1.31·23-s + 0.204·24-s + 2.30·25-s + 0.365·26-s + 0.962·27-s + 0.173·29-s − 0.742·30-s + 1.50·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 + 4.06T + 5T^{2} \) |
| 11 | \( 1 + 4.67T + 11T^{2} \) |
| 13 | \( 1 + 1.86T + 13T^{2} \) |
| 17 | \( 1 - 2.54T + 17T^{2} \) |
| 19 | \( 1 - 6.45T + 19T^{2} \) |
| 23 | \( 1 + 6.32T + 23T^{2} \) |
| 29 | \( 1 - 0.932T + 29T^{2} \) |
| 31 | \( 1 - 8.38T + 31T^{2} \) |
| 37 | \( 1 + 1.78T + 37T^{2} \) |
| 43 | \( 1 - 9.99T + 43T^{2} \) |
| 47 | \( 1 + 6.19T + 47T^{2} \) |
| 53 | \( 1 - 0.739T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 2.18T + 61T^{2} \) |
| 67 | \( 1 - 1.39T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 1.45T + 73T^{2} \) |
| 79 | \( 1 + 5.79T + 79T^{2} \) |
| 83 | \( 1 - 5.83T + 83T^{2} \) |
| 89 | \( 1 - 5.45T + 89T^{2} \) |
| 97 | \( 1 - 1.00T + 97T^{2} \) |
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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
−8.001056727311546092346075254498, −7.61970873998546400410789197785, −6.89405633447528392131875731804, −5.80999341501500385649457455283, −5.16171693057054394901433430381, −4.30278543217610876008074764955, −3.21800418201100706804047573273, −2.64645190648153209248337556541, −0.842866331575908248295970157323, 0,
0.842866331575908248295970157323, 2.64645190648153209248337556541, 3.21800418201100706804047573273, 4.30278543217610876008074764955, 5.16171693057054394901433430381, 5.80999341501500385649457455283, 6.89405633447528392131875731804, 7.61970873998546400410789197785, 8.001056727311546092346075254498
