| L(s) = 1 | − 1.46·2-s + 0.133·4-s + 0.593·5-s + 2.72·8-s − 0.866·10-s + 4.46·11-s + 4.51·13-s − 4.24·16-s + 0.273·17-s − 2.86·19-s + 0.0789·20-s − 6.51·22-s + 5.05·23-s − 4.64·25-s − 6.59·26-s + 0.352·29-s − 2.51·31-s + 0.751·32-s − 0.399·34-s − 6.64·37-s + 4.18·38-s + 1.61·40-s + 10.8·41-s + 3.38·43-s + 0.593·44-s − 7.38·46-s − 12.4·47-s + ⋯ |
| L(s) = 1 | − 1.03·2-s + 0.0665·4-s + 0.265·5-s + 0.964·8-s − 0.274·10-s + 1.34·11-s + 1.25·13-s − 1.06·16-s + 0.0662·17-s − 0.657·19-s + 0.0176·20-s − 1.38·22-s + 1.05·23-s − 0.929·25-s − 1.29·26-s + 0.0654·29-s − 0.451·31-s + 0.132·32-s − 0.0684·34-s − 1.09·37-s + 0.679·38-s + 0.255·40-s + 1.70·41-s + 0.515·43-s + 0.0894·44-s − 1.08·46-s − 1.81·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.058680303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.058680303\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.46T + 2T^{2} \) |
| 5 | \( 1 - 0.593T + 5T^{2} \) |
| 11 | \( 1 - 4.46T + 11T^{2} \) |
| 13 | \( 1 - 4.51T + 13T^{2} \) |
| 17 | \( 1 - 0.273T + 17T^{2} \) |
| 19 | \( 1 + 2.86T + 19T^{2} \) |
| 23 | \( 1 - 5.05T + 23T^{2} \) |
| 29 | \( 1 - 0.352T + 29T^{2} \) |
| 31 | \( 1 + 2.51T + 31T^{2} \) |
| 37 | \( 1 + 6.64T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 3.38T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 8.05T + 59T^{2} \) |
| 61 | \( 1 - 2.73T + 61T^{2} \) |
| 67 | \( 1 - 5.86T + 67T^{2} \) |
| 71 | \( 1 - 2.60T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 - 5.37T + 89T^{2} \) |
| 97 | \( 1 - 2.26T + 97T^{2} \) |
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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
−9.457251726913763622786642010357, −8.851823259636964720389022859657, −8.350549895640300295113138998835, −7.27386790827781380835338059794, −6.53388726348620445856529681002, −5.61088362510871379251986271949, −4.36256627475879219694345637827, −3.60221754580290317897134758412, −1.92347874271809749943724395961, −0.946329713497372231954577889354,
0.946329713497372231954577889354, 1.92347874271809749943724395961, 3.60221754580290317897134758412, 4.36256627475879219694345637827, 5.61088362510871379251986271949, 6.53388726348620445856529681002, 7.27386790827781380835338059794, 8.350549895640300295113138998835, 8.851823259636964720389022859657, 9.457251726913763622786642010357
