| L(s) = 1 | − 2.15·2-s − 2.27·3-s + 2.65·4-s + 4.91·6-s + 4.96·7-s − 1.41·8-s + 2.19·9-s + 11-s − 6.05·12-s + 7.07·13-s − 10.7·14-s − 2.25·16-s − 0.882·17-s − 4.72·18-s − 19-s − 11.3·21-s − 2.15·22-s − 0.450·23-s + 3.22·24-s − 15.2·26-s + 1.84·27-s + 13.1·28-s + 4.90·29-s − 4.59·31-s + 7.70·32-s − 2.27·33-s + 1.90·34-s + ⋯ |
| L(s) = 1 | − 1.52·2-s − 1.31·3-s + 1.32·4-s + 2.00·6-s + 1.87·7-s − 0.500·8-s + 0.730·9-s + 0.301·11-s − 1.74·12-s + 1.96·13-s − 2.86·14-s − 0.563·16-s − 0.214·17-s − 1.11·18-s − 0.229·19-s − 2.46·21-s − 0.460·22-s − 0.0940·23-s + 0.658·24-s − 2.99·26-s + 0.354·27-s + 2.49·28-s + 0.911·29-s − 0.824·31-s + 1.36·32-s − 0.396·33-s + 0.326·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8652455135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8652455135\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.15T + 2T^{2} \) |
| 3 | \( 1 + 2.27T + 3T^{2} \) |
| 7 | \( 1 - 4.96T + 7T^{2} \) |
| 13 | \( 1 - 7.07T + 13T^{2} \) |
| 17 | \( 1 + 0.882T + 17T^{2} \) |
| 23 | \( 1 + 0.450T + 23T^{2} \) |
| 29 | \( 1 - 4.90T + 29T^{2} \) |
| 31 | \( 1 + 4.59T + 31T^{2} \) |
| 37 | \( 1 - 9.57T + 37T^{2} \) |
| 41 | \( 1 + 3.73T + 41T^{2} \) |
| 43 | \( 1 - 6.29T + 43T^{2} \) |
| 47 | \( 1 + 5.17T + 47T^{2} \) |
| 53 | \( 1 + 3.00T + 53T^{2} \) |
| 59 | \( 1 - 7.98T + 59T^{2} \) |
| 61 | \( 1 - 2.63T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 - 6.34T + 79T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 89 | \( 1 - 5.27T + 89T^{2} \) |
| 97 | \( 1 + 19.4T + 97T^{2} \) |
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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
−8.314166966787445313528628910261, −7.75779683613153308441909450694, −6.78633402170708447883213916532, −6.24563887684653981219673509190, −5.43569725594745357630027518238, −4.68251960959837681843976660949, −3.91136367870102382442300053928, −2.22709309017810230238845094178, −1.30505817085820353106357034106, −0.821326783007062777735877436847,
0.821326783007062777735877436847, 1.30505817085820353106357034106, 2.22709309017810230238845094178, 3.91136367870102382442300053928, 4.68251960959837681843976660949, 5.43569725594745357630027518238, 6.24563887684653981219673509190, 6.78633402170708447883213916532, 7.75779683613153308441909450694, 8.314166966787445313528628910261
