| L(s) = 1 | + 2.45·2-s − 3.25·3-s + 4.02·4-s + 1.44·5-s − 7.98·6-s + 0.736·7-s + 4.96·8-s + 7.60·9-s + 3.53·10-s − 0.631·11-s − 13.0·12-s + 0.659·13-s + 1.80·14-s − 4.69·15-s + 4.12·16-s + 3.60·17-s + 18.6·18-s + 3.07·19-s + 5.80·20-s − 2.39·21-s − 1.54·22-s + 7.24·23-s − 16.1·24-s − 2.91·25-s + 1.61·26-s − 14.9·27-s + 2.96·28-s + ⋯ |
| L(s) = 1 | + 1.73·2-s − 1.87·3-s + 2.01·4-s + 0.645·5-s − 3.26·6-s + 0.278·7-s + 1.75·8-s + 2.53·9-s + 1.11·10-s − 0.190·11-s − 3.77·12-s + 0.182·13-s + 0.483·14-s − 1.21·15-s + 1.03·16-s + 0.873·17-s + 4.39·18-s + 0.704·19-s + 1.29·20-s − 0.523·21-s − 0.330·22-s + 1.51·23-s − 3.29·24-s − 0.583·25-s + 0.317·26-s − 2.88·27-s + 0.559·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.559953790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.559953790\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 503 | \( 1 - T \) |
| good | 2 | \( 1 - 2.45T + 2T^{2} \) |
| 3 | \( 1 + 3.25T + 3T^{2} \) |
| 5 | \( 1 - 1.44T + 5T^{2} \) |
| 7 | \( 1 - 0.736T + 7T^{2} \) |
| 11 | \( 1 + 0.631T + 11T^{2} \) |
| 13 | \( 1 - 0.659T + 13T^{2} \) |
| 17 | \( 1 - 3.60T + 17T^{2} \) |
| 19 | \( 1 - 3.07T + 19T^{2} \) |
| 23 | \( 1 - 7.24T + 23T^{2} \) |
| 29 | \( 1 - 4.33T + 29T^{2} \) |
| 31 | \( 1 + 4.66T + 31T^{2} \) |
| 37 | \( 1 - 8.87T + 37T^{2} \) |
| 41 | \( 1 + 8.77T + 41T^{2} \) |
| 43 | \( 1 + 5.16T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 0.0719T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 + 6.02T + 61T^{2} \) |
| 67 | \( 1 + 7.19T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 9.16T + 73T^{2} \) |
| 79 | \( 1 + 1.92T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 - 9.56T + 97T^{2} \) |
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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
−11.36253107947850370441085337066, −10.52363554187209540839574622529, −9.639249098652850666853884052668, −7.59688164132940206613037320799, −6.61456021722322862036298933501, −5.99556625858581627899249765779, −5.16075234457117220106193225689, −4.76481274877360702754328148638, −3.33020513051775815744069076209, −1.51059181377813626025205905238,
1.51059181377813626025205905238, 3.33020513051775815744069076209, 4.76481274877360702754328148638, 5.16075234457117220106193225689, 5.99556625858581627899249765779, 6.61456021722322862036298933501, 7.59688164132940206613037320799, 9.639249098652850666853884052668, 10.52363554187209540839574622529, 11.36253107947850370441085337066
