| L(s) = 1 | − 2.20·2-s + 2.87·4-s + 3.80·5-s − 1.93·8-s − 8.40·10-s − 4.32·11-s − 2.87·13-s − 1.48·16-s − 4.02·17-s − 1.60·19-s + 10.9·20-s + 9.54·22-s + 2.66·23-s + 9.49·25-s + 6.34·26-s − 0.750·29-s + 0.140·31-s + 7.14·32-s + 8.88·34-s − 8.28·37-s + 3.55·38-s − 7.35·40-s + 10.3·41-s + 0.267·43-s − 12.4·44-s − 5.88·46-s + 7.93·47-s + ⋯ |
| L(s) = 1 | − 1.56·2-s + 1.43·4-s + 1.70·5-s − 0.683·8-s − 2.65·10-s − 1.30·11-s − 0.797·13-s − 0.370·16-s − 0.976·17-s − 0.368·19-s + 2.44·20-s + 2.03·22-s + 0.556·23-s + 1.89·25-s + 1.24·26-s − 0.139·29-s + 0.0252·31-s + 1.26·32-s + 1.52·34-s − 1.36·37-s + 0.576·38-s − 1.16·40-s + 1.62·41-s + 0.0407·43-s − 1.87·44-s − 0.868·46-s + 1.15·47-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.20T + 2T^{2} \) |
| 5 | \( 1 - 3.80T + 5T^{2} \) |
| 11 | \( 1 + 4.32T + 11T^{2} \) |
| 13 | \( 1 + 2.87T + 13T^{2} \) |
| 17 | \( 1 + 4.02T + 17T^{2} \) |
| 19 | \( 1 + 1.60T + 19T^{2} \) |
| 23 | \( 1 - 2.66T + 23T^{2} \) |
| 29 | \( 1 + 0.750T + 29T^{2} \) |
| 31 | \( 1 - 0.140T + 31T^{2} \) |
| 37 | \( 1 + 8.28T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 0.267T + 43T^{2} \) |
| 47 | \( 1 - 7.93T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 0.693T + 59T^{2} \) |
| 61 | \( 1 - 2.10T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 3.62T + 71T^{2} \) |
| 73 | \( 1 + 3.57T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 6.44T + 83T^{2} \) |
| 89 | \( 1 + 0.256T + 89T^{2} \) |
| 97 | \( 1 - 1.05T + 97T^{2} \) |
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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
−8.355433965106544757724370537040, −7.33125467302904001834149808773, −6.96866036996444368871344580926, −5.97082052995654945288800583142, −5.34698656975078951580029466777, −4.46226074337265436859034769152, −2.58207872135163002831511192968, −2.39717914282193320310745306059, −1.35099397022673191617344145123, 0,
1.35099397022673191617344145123, 2.39717914282193320310745306059, 2.58207872135163002831511192968, 4.46226074337265436859034769152, 5.34698656975078951580029466777, 5.97082052995654945288800583142, 6.96866036996444368871344580926, 7.33125467302904001834149808773, 8.355433965106544757724370537040
