| L(s) = 1 | − 2.53·2-s + 4.41·4-s + 1.34·5-s + 1.53·7-s − 6.10·8-s − 3.41·10-s + 1.18·11-s − 2.71·13-s − 3.87·14-s + 6.63·16-s − 3.87·17-s + 5.94·20-s − 3.00·22-s + 5.06·23-s − 3.18·25-s + 6.87·26-s + 6.75·28-s − 4.65·29-s + 3.83·31-s − 4.59·32-s + 9.82·34-s + 2.06·35-s + 4.10·37-s − 8.22·40-s − 9.98·41-s − 8.70·43-s + 5.22·44-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 2.20·4-s + 0.602·5-s + 0.579·7-s − 2.15·8-s − 1.07·10-s + 0.357·11-s − 0.753·13-s − 1.03·14-s + 1.65·16-s − 0.940·17-s + 1.32·20-s − 0.639·22-s + 1.05·23-s − 0.636·25-s + 1.34·26-s + 1.27·28-s − 0.863·29-s + 0.689·31-s − 0.812·32-s + 1.68·34-s + 0.348·35-s + 0.675·37-s − 1.30·40-s − 1.55·41-s − 1.32·43-s + 0.787·44-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 2.53T + 2T^{2} \) |
| 5 | \( 1 - 1.34T + 5T^{2} \) |
| 7 | \( 1 - 1.53T + 7T^{2} \) |
| 11 | \( 1 - 1.18T + 11T^{2} \) |
| 13 | \( 1 + 2.71T + 13T^{2} \) |
| 17 | \( 1 + 3.87T + 17T^{2} \) |
| 23 | \( 1 - 5.06T + 23T^{2} \) |
| 29 | \( 1 + 4.65T + 29T^{2} \) |
| 31 | \( 1 - 3.83T + 31T^{2} \) |
| 37 | \( 1 - 4.10T + 37T^{2} \) |
| 41 | \( 1 + 9.98T + 41T^{2} \) |
| 43 | \( 1 + 8.70T + 43T^{2} \) |
| 47 | \( 1 + 0.573T + 47T^{2} \) |
| 53 | \( 1 - 2.94T + 53T^{2} \) |
| 59 | \( 1 + 3.93T + 59T^{2} \) |
| 61 | \( 1 + 4.51T + 61T^{2} \) |
| 67 | \( 1 - 3.88T + 67T^{2} \) |
| 71 | \( 1 + 6.93T + 71T^{2} \) |
| 73 | \( 1 - 6.12T + 73T^{2} \) |
| 79 | \( 1 + 9.80T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 + 2.42T + 89T^{2} \) |
| 97 | \( 1 + 7.36T + 97T^{2} \) |
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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
−8.443093710280267872241745292309, −7.72030734271086245669972859197, −6.96009124499980283078293197402, −6.44192460960282242635161023644, −5.41381228025841099546152283420, −4.47979811148684248884135718600, −3.02364341133847395127289265924, −2.07119106061965853642348112479, −1.41622820564335959410394786195, 0,
1.41622820564335959410394786195, 2.07119106061965853642348112479, 3.02364341133847395127289265924, 4.47979811148684248884135718600, 5.41381228025841099546152283420, 6.44192460960282242635161023644, 6.96009124499980283078293197402, 7.72030734271086245669972859197, 8.443093710280267872241745292309
