| L(s) = 1 | + 2.47·2-s − 1.11·3-s + 4.11·4-s − 0.357·5-s − 2.75·6-s − 7-s + 5.22·8-s − 1.75·9-s − 0.885·10-s + 0.527·11-s − 4.58·12-s − 0.357·13-s − 2.47·14-s + 0.399·15-s + 4.70·16-s − 3.70·17-s − 4.34·18-s + 19-s − 1.47·20-s + 1.11·21-s + 1.30·22-s + 8.58·23-s − 5.83·24-s − 4.87·25-s − 0.885·26-s + 5.30·27-s − 4.11·28-s + ⋯ |
| L(s) = 1 | + 1.74·2-s − 0.643·3-s + 2.05·4-s − 0.160·5-s − 1.12·6-s − 0.377·7-s + 1.84·8-s − 0.585·9-s − 0.279·10-s + 0.158·11-s − 1.32·12-s − 0.0992·13-s − 0.660·14-s + 0.103·15-s + 1.17·16-s − 0.898·17-s − 1.02·18-s + 0.229·19-s − 0.329·20-s + 0.243·21-s + 0.277·22-s + 1.79·23-s − 1.19·24-s − 0.974·25-s − 0.173·26-s + 1.02·27-s − 0.777·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.040730021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.040730021\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.47T + 2T^{2} \) |
| 3 | \( 1 + 1.11T + 3T^{2} \) |
| 5 | \( 1 + 0.357T + 5T^{2} \) |
| 11 | \( 1 - 0.527T + 11T^{2} \) |
| 13 | \( 1 + 0.357T + 13T^{2} \) |
| 17 | \( 1 + 3.70T + 17T^{2} \) |
| 23 | \( 1 - 8.58T + 23T^{2} \) |
| 29 | \( 1 + 4.04T + 29T^{2} \) |
| 31 | \( 1 + 1.24T + 31T^{2} \) |
| 37 | \( 1 - 3.30T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 + 0.715T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 6.06T + 53T^{2} \) |
| 59 | \( 1 + 7.87T + 59T^{2} \) |
| 61 | \( 1 - 5.53T + 61T^{2} \) |
| 67 | \( 1 - 8.64T + 67T^{2} \) |
| 71 | \( 1 + 7.53T + 71T^{2} \) |
| 73 | \( 1 + 9.00T + 73T^{2} \) |
| 79 | \( 1 + 9.17T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 9.07T + 97T^{2} \) |
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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
−13.18031241420986874233485744161, −12.44386657504700461336205289193, −11.40930499912040394556741286844, −10.92698204225239212074827022861, −9.149906346979170698503018299771, −7.31479207275723987321850489043, −6.24351851253870846067773737237, −5.38271752009440049613109356381, −4.19170633252969644278020354639, −2.78451081694730637976123964525,
2.78451081694730637976123964525, 4.19170633252969644278020354639, 5.38271752009440049613109356381, 6.24351851253870846067773737237, 7.31479207275723987321850489043, 9.149906346979170698503018299771, 10.92698204225239212074827022861, 11.40930499912040394556741286844, 12.44386657504700461336205289193, 13.18031241420986874233485744161
