| L(s) = 1 | + 2.66·2-s − 3-s + 5.12·4-s − 2.66·6-s − 0.454·7-s + 8.33·8-s + 9-s − 1.45·11-s − 5.12·12-s + 3.33·13-s − 1.21·14-s + 12.0·16-s + 2.66·18-s + 19-s + 0.454·21-s − 3.88·22-s + 6.79·23-s − 8.33·24-s + 8.90·26-s − 27-s − 2.33·28-s + 3·29-s + 8.79·31-s + 15.3·32-s + 1.45·33-s + 5.12·36-s − 10.2·37-s + ⋯ |
| L(s) = 1 | + 1.88·2-s − 0.577·3-s + 2.56·4-s − 1.08·6-s − 0.171·7-s + 2.94·8-s + 0.333·9-s − 0.438·11-s − 1.47·12-s + 0.925·13-s − 0.324·14-s + 3.00·16-s + 0.629·18-s + 0.229·19-s + 0.0992·21-s − 0.827·22-s + 1.41·23-s − 1.70·24-s + 1.74·26-s − 0.192·27-s − 0.440·28-s + 0.557·29-s + 1.57·31-s + 2.71·32-s + 0.253·33-s + 0.853·36-s − 1.68·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.766548459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.766548459\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.66T + 2T^{2} \) |
| 7 | \( 1 + 0.454T + 7T^{2} \) |
| 11 | \( 1 + 1.45T + 11T^{2} \) |
| 13 | \( 1 - 3.33T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 - 6.79T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 8.79T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 3.97T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 2.42T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 - 7.54T + 59T^{2} \) |
| 61 | \( 1 + 3.90T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 + 9.13T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 8.97T + 79T^{2} \) |
| 83 | \( 1 - 4.54T + 83T^{2} \) |
| 89 | \( 1 + 0.0901T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−9.925589900630465361585039426509, −8.549468594476719174902879978824, −7.51195848947744866948043896603, −6.58983323319649840046065785904, −6.19967489982773101542946304391, −5.12702968893738158345610412971, −4.73456259989932119361860466904, −3.56290585552489208401110399726, −2.86595049763457783970293569030, −1.44395186376988408355400029909,
1.44395186376988408355400029909, 2.86595049763457783970293569030, 3.56290585552489208401110399726, 4.73456259989932119361860466904, 5.12702968893738158345610412971, 6.19967489982773101542946304391, 6.58983323319649840046065785904, 7.51195848947744866948043896603, 8.549468594476719174902879978824, 9.925589900630465361585039426509
