| L(s) = 1 | + 0.00234·2-s + 2.44·3-s − 1.99·4-s + 1.45·5-s + 0.00574·6-s + 4.30·7-s − 0.00939·8-s + 2.97·9-s + 0.00342·10-s − 11-s − 4.89·12-s + 6.13·13-s + 0.0101·14-s + 3.56·15-s + 3.99·16-s + 3.35·17-s + 0.00699·18-s − 4.42·19-s − 2.91·20-s + 10.5·21-s − 0.00234·22-s − 6.97·23-s − 0.0229·24-s − 2.87·25-s + 0.0143·26-s − 0.0502·27-s − 8.60·28-s + ⋯ |
| L(s) = 1 | + 0.00165·2-s + 1.41·3-s − 0.999·4-s + 0.652·5-s + 0.00234·6-s + 1.62·7-s − 0.00331·8-s + 0.993·9-s + 0.00108·10-s − 0.301·11-s − 1.41·12-s + 1.70·13-s + 0.00270·14-s + 0.921·15-s + 0.999·16-s + 0.813·17-s + 0.00164·18-s − 1.01·19-s − 0.652·20-s + 2.29·21-s − 0.000500·22-s − 1.45·23-s − 0.00468·24-s − 0.574·25-s + 0.00282·26-s − 0.00967·27-s − 1.62·28-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
| \(L(1)\) |
\(\approx\) |
\(3.008642848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.008642848\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.00234T + 2T^{2} \) |
| 3 | \( 1 - 2.44T + 3T^{2} \) |
| 5 | \( 1 - 1.45T + 5T^{2} \) |
| 7 | \( 1 - 4.30T + 7T^{2} \) |
| 13 | \( 1 - 6.13T + 13T^{2} \) |
| 17 | \( 1 - 3.35T + 17T^{2} \) |
| 19 | \( 1 + 4.42T + 19T^{2} \) |
| 23 | \( 1 + 6.97T + 23T^{2} \) |
| 29 | \( 1 - 1.53T + 29T^{2} \) |
| 31 | \( 1 + 2.56T + 31T^{2} \) |
| 37 | \( 1 + 8.05T + 37T^{2} \) |
| 41 | \( 1 + 0.972T + 41T^{2} \) |
| 43 | \( 1 - 7.31T + 43T^{2} \) |
| 47 | \( 1 - 8.14T + 47T^{2} \) |
| 53 | \( 1 - 2.35T + 53T^{2} \) |
| 59 | \( 1 - 5.79T + 59T^{2} \) |
| 61 | \( 1 + 8.35T + 61T^{2} \) |
| 67 | \( 1 + 7.08T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 - 0.310T + 73T^{2} \) |
| 79 | \( 1 - 1.50T + 79T^{2} \) |
| 83 | \( 1 + 5.64T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 + 6.91T + 97T^{2} \) |
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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
−9.204134301328472887882279010896, −8.662778846810835893943010541535, −8.142368493939089837125338659009, −7.65426513677210323608671294123, −6.02950961621615257903910120529, −5.33551160812834977896605847930, −4.17000840077191008200207472231, −3.66965381945900555558901501208, −2.22026240190339374730959851244, −1.40871107444661502452458001087,
1.40871107444661502452458001087, 2.22026240190339374730959851244, 3.66965381945900555558901501208, 4.17000840077191008200207472231, 5.33551160812834977896605847930, 6.02950961621615257903910120529, 7.65426513677210323608671294123, 8.142368493939089837125338659009, 8.662778846810835893943010541535, 9.204134301328472887882279010896
