L(s) = 1 | − 1.18·2-s − 0.345·3-s − 0.590·4-s + 0.409·6-s − 2.02·7-s + 3.07·8-s − 2.88·9-s + 3.88·11-s + 0.203·12-s + 2.40·14-s − 2.47·16-s + 5.45·17-s + 3.42·18-s + 5.88·19-s + 0.700·21-s − 4.60·22-s + 0.345·23-s − 1.06·24-s + 2.02·27-s + 1.19·28-s + 3·29-s + 1.18·31-s − 3.21·32-s − 1.33·33-s − 6.47·34-s + 1.70·36-s − 5.45·37-s + ⋯ |
L(s) = 1 | − 0.839·2-s − 0.199·3-s − 0.295·4-s + 0.167·6-s − 0.767·7-s + 1.08·8-s − 0.960·9-s + 1.17·11-s + 0.0588·12-s + 0.644·14-s − 0.617·16-s + 1.32·17-s + 0.806·18-s + 1.34·19-s + 0.152·21-s − 0.982·22-s + 0.0719·23-s − 0.216·24-s + 0.390·27-s + 0.226·28-s + 0.557·29-s + 0.212·31-s − 0.568·32-s − 0.233·33-s − 1.10·34-s + 0.283·36-s − 0.895·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8345031972\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8345031972\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.18T + 2T^{2} \) |
| 3 | \( 1 + 0.345T + 3T^{2} \) |
| 7 | \( 1 + 2.02T + 7T^{2} \) |
| 11 | \( 1 - 3.88T + 11T^{2} \) |
| 17 | \( 1 - 5.45T + 17T^{2} \) |
| 19 | \( 1 - 5.88T + 19T^{2} \) |
| 23 | \( 1 - 0.345T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 1.18T + 31T^{2} \) |
| 37 | \( 1 + 5.45T + 37T^{2} \) |
| 41 | \( 1 - 0.180T + 41T^{2} \) |
| 43 | \( 1 - 1.33T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 + 2.42T + 53T^{2} \) |
| 59 | \( 1 + 7.06T + 59T^{2} \) |
| 61 | \( 1 - 6.76T + 61T^{2} \) |
| 67 | \( 1 - 4.40T + 67T^{2} \) |
| 71 | \( 1 + 1.88T + 71T^{2} \) |
| 73 | \( 1 + 8.86T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 7.83T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 - 5.80T + 97T^{2} \) |
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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
−8.459215529730271561560377997522, −7.86055442622780297668888488078, −7.02528679970111038455988583647, −6.29968831998547436453979593768, −5.48417063959919873466555109627, −4.76032497481205879196873528676, −3.58645696898119085117370171771, −3.09620557087689456219660089266, −1.54463142643230521070862371139, −0.63728415183471449482695562756,
0.63728415183471449482695562756, 1.54463142643230521070862371139, 3.09620557087689456219660089266, 3.58645696898119085117370171771, 4.76032497481205879196873528676, 5.48417063959919873466555109627, 6.29968831998547436453979593768, 7.02528679970111038455988583647, 7.86055442622780297668888488078, 8.459215529730271561560377997522