| L(s) = 1 | + 0.571·2-s − 1.67·4-s + 2.67·5-s − 3.67·7-s − 2.10·8-s + 1.52·10-s + 3.81·11-s + 0.143·13-s − 2.10·14-s + 2.14·16-s − 4.47·20-s + 2.18·22-s − 7.52·23-s + 2.14·25-s + 0.0823·26-s + 6.14·28-s + 5.34·29-s + 8.81·31-s + 5.42·32-s − 9.81·35-s − 37-s − 5.61·40-s − 5.34·41-s + 2.81·43-s − 6.38·44-s − 4.30·46-s + 6·47-s + ⋯ |
| L(s) = 1 | + 0.404·2-s − 0.836·4-s + 1.19·5-s − 1.38·7-s − 0.742·8-s + 0.483·10-s + 1.15·11-s + 0.0399·13-s − 0.561·14-s + 0.535·16-s − 0.999·20-s + 0.465·22-s − 1.56·23-s + 0.428·25-s + 0.0161·26-s + 1.16·28-s + 0.992·29-s + 1.58·31-s + 0.959·32-s − 1.65·35-s − 0.164·37-s − 0.887·40-s − 0.834·41-s + 0.429·43-s − 0.962·44-s − 0.634·46-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.894700718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.894700718\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 0.571T + 2T^{2} \) |
| 5 | \( 1 - 2.67T + 5T^{2} \) |
| 7 | \( 1 + 3.67T + 7T^{2} \) |
| 11 | \( 1 - 3.81T + 11T^{2} \) |
| 13 | \( 1 - 0.143T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 7.52T + 23T^{2} \) |
| 29 | \( 1 - 5.34T + 29T^{2} \) |
| 31 | \( 1 - 8.81T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + 5.34T + 41T^{2} \) |
| 43 | \( 1 - 2.81T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 8.01T + 53T^{2} \) |
| 59 | \( 1 + 3.81T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 - 5.38T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 0.345T + 73T^{2} \) |
| 79 | \( 1 - 6.52T + 79T^{2} \) |
| 83 | \( 1 - 2.28T + 83T^{2} \) |
| 89 | \( 1 - 8.67T + 89T^{2} \) |
| 97 | \( 1 + 5.91T + 97T^{2} \) |
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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
−8.819699861535652333457651229901, −8.103197768413114787782506481868, −6.71592137314820929266741566390, −6.28262575387051116509695506992, −5.79815328535240558271846677929, −4.78693825267277723977174248434, −3.92867131592538724401952333760, −3.21822089476621381776739753226, −2.16246149303960155526409308322, −0.78156480231356303898854791481,
0.78156480231356303898854791481, 2.16246149303960155526409308322, 3.21822089476621381776739753226, 3.92867131592538724401952333760, 4.78693825267277723977174248434, 5.79815328535240558271846677929, 6.28262575387051116509695506992, 6.71592137314820929266741566390, 8.103197768413114787782506481868, 8.819699861535652333457651229901
