| L(s) = 1 | − 2.17·2-s + 2.73·4-s − 1.26·5-s − 1.60·8-s + 2.76·10-s + 5.47·11-s − 4.74·13-s − 1.98·16-s + 4.81·17-s + 5.38·19-s − 3.47·20-s − 11.9·22-s + 5.17·23-s − 3.39·25-s + 10.3·26-s + 4.03·29-s − 1.46·31-s + 7.52·32-s − 10.4·34-s + 1.91·37-s − 11.7·38-s + 2.03·40-s − 3.89·41-s + 3.32·43-s + 14.9·44-s − 11.2·46-s − 3.15·47-s + ⋯ |
| L(s) = 1 | − 1.53·2-s + 1.36·4-s − 0.567·5-s − 0.566·8-s + 0.872·10-s + 1.65·11-s − 1.31·13-s − 0.496·16-s + 1.16·17-s + 1.23·19-s − 0.776·20-s − 2.54·22-s + 1.07·23-s − 0.678·25-s + 2.02·26-s + 0.748·29-s − 0.262·31-s + 1.33·32-s − 1.79·34-s + 0.315·37-s − 1.89·38-s + 0.321·40-s − 0.608·41-s + 0.506·43-s + 2.25·44-s − 1.66·46-s − 0.460·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8477043301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8477043301\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.17T + 2T^{2} \) |
| 5 | \( 1 + 1.26T + 5T^{2} \) |
| 11 | \( 1 - 5.47T + 11T^{2} \) |
| 13 | \( 1 + 4.74T + 13T^{2} \) |
| 17 | \( 1 - 4.81T + 17T^{2} \) |
| 19 | \( 1 - 5.38T + 19T^{2} \) |
| 23 | \( 1 - 5.17T + 23T^{2} \) |
| 29 | \( 1 - 4.03T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 - 1.91T + 37T^{2} \) |
| 41 | \( 1 + 3.89T + 41T^{2} \) |
| 43 | \( 1 - 3.32T + 43T^{2} \) |
| 47 | \( 1 + 3.15T + 47T^{2} \) |
| 53 | \( 1 - 7.14T + 53T^{2} \) |
| 59 | \( 1 + 0.308T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 4.47T + 67T^{2} \) |
| 71 | \( 1 - 1.96T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 9.01T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 - 4.97T + 97T^{2} \) |
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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
−8.567109440854082894615023370676, −7.72826356625367028644220576546, −7.24917351176622642750939472083, −6.73567570262907303336763382600, −5.60870890959715668528580522297, −4.65942864945110429594120352907, −3.70384615600723674291589186792, −2.75046923357183204584141206332, −1.50615990554576602898802261822, −0.72769623232698717217168294911,
0.72769623232698717217168294911, 1.50615990554576602898802261822, 2.75046923357183204584141206332, 3.70384615600723674291589186792, 4.65942864945110429594120352907, 5.60870890959715668528580522297, 6.73567570262907303336763382600, 7.24917351176622642750939472083, 7.72826356625367028644220576546, 8.567109440854082894615023370676
