| L(s) = 1 | + 2.15·2-s − 2.82·3-s + 2.65·4-s + 0.00102·5-s − 6.10·6-s − 0.810·7-s + 1.41·8-s + 5.00·9-s + 0.00220·10-s − 11-s − 7.50·12-s + 2.59·13-s − 1.74·14-s − 0.00289·15-s − 2.26·16-s + 1.35·17-s + 10.7·18-s − 3.40·19-s + 0.00271·20-s + 2.29·21-s − 2.15·22-s − 3.47·23-s − 3.99·24-s − 4.99·25-s + 5.60·26-s − 5.65·27-s − 2.15·28-s + ⋯ |
| L(s) = 1 | + 1.52·2-s − 1.63·3-s + 1.32·4-s + 0.000457·5-s − 2.49·6-s − 0.306·7-s + 0.499·8-s + 1.66·9-s + 0.000698·10-s − 0.301·11-s − 2.16·12-s + 0.720·13-s − 0.467·14-s − 0.000747·15-s − 0.565·16-s + 0.327·17-s + 2.54·18-s − 0.781·19-s + 0.000607·20-s + 0.500·21-s − 0.459·22-s − 0.723·23-s − 0.815·24-s − 0.999·25-s + 1.09·26-s − 1.08·27-s − 0.406·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 - 2.15T + 2T^{2} \) |
| 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 - 0.00102T + 5T^{2} \) |
| 7 | \( 1 + 0.810T + 7T^{2} \) |
| 13 | \( 1 - 2.59T + 13T^{2} \) |
| 17 | \( 1 - 1.35T + 17T^{2} \) |
| 19 | \( 1 + 3.40T + 19T^{2} \) |
| 23 | \( 1 + 3.47T + 23T^{2} \) |
| 29 | \( 1 + 0.561T + 29T^{2} \) |
| 31 | \( 1 + 8.77T + 31T^{2} \) |
| 37 | \( 1 - 3.41T + 37T^{2} \) |
| 41 | \( 1 - 3.80T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 + 2.55T + 47T^{2} \) |
| 53 | \( 1 + 8.64T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 + 9.43T + 61T^{2} \) |
| 67 | \( 1 - 5.99T + 67T^{2} \) |
| 71 | \( 1 + 8.12T + 71T^{2} \) |
| 73 | \( 1 + 0.305T + 73T^{2} \) |
| 79 | \( 1 - 6.69T + 79T^{2} \) |
| 83 | \( 1 + 8.01T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−9.376304329308669677619451033671, −8.038908489366562358829732747388, −6.94488906641128621567122840473, −6.20525279183066550421431696919, −5.79643135347323297092172561021, −5.03616790485083436674326431834, −4.20611385149697751442791799717, −3.40363581403669144521131659901, −1.86844992131300190422345351576, 0,
1.86844992131300190422345351576, 3.40363581403669144521131659901, 4.20611385149697751442791799717, 5.03616790485083436674326431834, 5.79643135347323297092172561021, 6.20525279183066550421431696919, 6.94488906641128621567122840473, 8.038908489366562358829732747388, 9.376304329308669677619451033671
