| L(s) = 1 | − 2.23·3-s − 1.70·5-s + 1.97·9-s + 5.50·11-s − 0.699·13-s + 3.80·15-s − 0.729·17-s + 6.13·19-s + 7.80·23-s − 2.09·25-s + 2.28·27-s + 8.42·29-s + 7.93·31-s − 12.2·33-s − 6.83·37-s + 1.56·39-s + 41-s + 10.3·43-s − 3.36·45-s − 1.23·47-s + 1.62·51-s − 2.32·53-s − 9.38·55-s − 13.6·57-s − 5.25·59-s − 9.64·61-s + 1.19·65-s + ⋯ |
| L(s) = 1 | − 1.28·3-s − 0.762·5-s + 0.658·9-s + 1.65·11-s − 0.194·13-s + 0.981·15-s − 0.176·17-s + 1.40·19-s + 1.62·23-s − 0.419·25-s + 0.440·27-s + 1.56·29-s + 1.42·31-s − 2.13·33-s − 1.12·37-s + 0.249·39-s + 0.156·41-s + 1.57·43-s − 0.501·45-s − 0.179·47-s + 0.227·51-s − 0.319·53-s − 1.26·55-s − 1.81·57-s − 0.683·59-s − 1.23·61-s + 0.147·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.315161765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.315161765\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 5 | \( 1 + 1.70T + 5T^{2} \) |
| 11 | \( 1 - 5.50T + 11T^{2} \) |
| 13 | \( 1 + 0.699T + 13T^{2} \) |
| 17 | \( 1 + 0.729T + 17T^{2} \) |
| 19 | \( 1 - 6.13T + 19T^{2} \) |
| 23 | \( 1 - 7.80T + 23T^{2} \) |
| 29 | \( 1 - 8.42T + 29T^{2} \) |
| 31 | \( 1 - 7.93T + 31T^{2} \) |
| 37 | \( 1 + 6.83T + 37T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 1.23T + 47T^{2} \) |
| 53 | \( 1 + 2.32T + 53T^{2} \) |
| 59 | \( 1 + 5.25T + 59T^{2} \) |
| 61 | \( 1 + 9.64T + 61T^{2} \) |
| 67 | \( 1 + 1.85T + 67T^{2} \) |
| 71 | \( 1 + 8.36T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 - 7.15T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 2.20T + 89T^{2} \) |
| 97 | \( 1 - 9.07T + 97T^{2} \) |
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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
−7.66217765195531191299059914462, −6.93850700565181004226795461689, −6.50153611613535012431801925828, −5.82975067695509245909486980655, −4.90645580344127310918731146194, −4.54048612284391477483132911546, −3.60524026200705869037703900087, −2.84516293104660506791466350955, −1.27805304325506079431114191425, −0.71842876382335821841203219001,
0.71842876382335821841203219001, 1.27805304325506079431114191425, 2.84516293104660506791466350955, 3.60524026200705869037703900087, 4.54048612284391477483132911546, 4.90645580344127310918731146194, 5.82975067695509245909486980655, 6.50153611613535012431801925828, 6.93850700565181004226795461689, 7.66217765195531191299059914462
