| L(s) = 1 | − 1.47·2-s + 1.55·3-s + 0.162·4-s − 4.37·5-s − 2.29·6-s + 2.70·8-s − 0.568·9-s + 6.42·10-s + 0.334·11-s + 0.253·12-s − 5.80·13-s − 6.81·15-s − 4.29·16-s − 2.86·17-s + 0.835·18-s − 0.421·19-s − 0.710·20-s − 0.491·22-s + 2.00·23-s + 4.21·24-s + 14.1·25-s + 8.54·26-s − 5.56·27-s + 4.99·29-s + 10.0·30-s − 31-s + 0.917·32-s + ⋯ |
| L(s) = 1 | − 1.03·2-s + 0.900·3-s + 0.0812·4-s − 1.95·5-s − 0.936·6-s + 0.955·8-s − 0.189·9-s + 2.03·10-s + 0.100·11-s + 0.0731·12-s − 1.61·13-s − 1.76·15-s − 1.07·16-s − 0.695·17-s + 0.196·18-s − 0.0966·19-s − 0.158·20-s − 0.104·22-s + 0.418·23-s + 0.860·24-s + 2.82·25-s + 1.67·26-s − 1.07·27-s + 0.927·29-s + 1.83·30-s − 0.179·31-s + 0.162·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4675683708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4675683708\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.47T + 2T^{2} \) |
| 3 | \( 1 - 1.55T + 3T^{2} \) |
| 5 | \( 1 + 4.37T + 5T^{2} \) |
| 11 | \( 1 - 0.334T + 11T^{2} \) |
| 13 | \( 1 + 5.80T + 13T^{2} \) |
| 17 | \( 1 + 2.86T + 17T^{2} \) |
| 19 | \( 1 + 0.421T + 19T^{2} \) |
| 23 | \( 1 - 2.00T + 23T^{2} \) |
| 29 | \( 1 - 4.99T + 29T^{2} \) |
| 37 | \( 1 + 4.42T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 4.91T + 43T^{2} \) |
| 47 | \( 1 - 1.20T + 47T^{2} \) |
| 53 | \( 1 + 5.10T + 53T^{2} \) |
| 59 | \( 1 - 7.94T + 59T^{2} \) |
| 61 | \( 1 - 1.02T + 61T^{2} \) |
| 67 | \( 1 - 7.16T + 67T^{2} \) |
| 71 | \( 1 - 1.51T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 7.74T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 2.43T + 89T^{2} \) |
| 97 | \( 1 + 4.26T + 97T^{2} \) |
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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
−9.145139230665083628607637811540, −8.644106192419141770057205243948, −7.990906195268793386324740748044, −7.47512004505035978800098082035, −6.84628584576950346133587200213, −4.95842840716541890208886946912, −4.32304196401218716104501164273, −3.35969094898382847429988554733, −2.33951535895322817159852730273, −0.51981465329015380446130768462,
0.51981465329015380446130768462, 2.33951535895322817159852730273, 3.35969094898382847429988554733, 4.32304196401218716104501164273, 4.95842840716541890208886946912, 6.84628584576950346133587200213, 7.47512004505035978800098082035, 7.990906195268793386324740748044, 8.644106192419141770057205243948, 9.145139230665083628607637811540
