| L(s) = 1 | + 0.101·2-s − 0.700·3-s − 1.98·4-s + 0.497·5-s − 0.0708·6-s − 5.07·7-s − 0.403·8-s − 2.50·9-s + 0.0504·10-s − 0.805·11-s + 1.39·12-s + 1.59·13-s − 0.513·14-s − 0.348·15-s + 3.93·16-s + 17-s − 0.254·18-s + 3.21·19-s − 0.990·20-s + 3.55·21-s − 0.0815·22-s + 0.725·23-s + 0.282·24-s − 4.75·25-s + 0.161·26-s + 3.85·27-s + 10.1·28-s + ⋯ |
| L(s) = 1 | + 0.0715·2-s − 0.404·3-s − 0.994·4-s + 0.222·5-s − 0.0289·6-s − 1.91·7-s − 0.142·8-s − 0.836·9-s + 0.0159·10-s − 0.242·11-s + 0.402·12-s + 0.442·13-s − 0.137·14-s − 0.0900·15-s + 0.984·16-s + 0.242·17-s − 0.0598·18-s + 0.738·19-s − 0.221·20-s + 0.775·21-s − 0.0173·22-s + 0.151·23-s + 0.0577·24-s − 0.950·25-s + 0.0317·26-s + 0.742·27-s + 1.90·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6488551887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6488551887\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 - 0.101T + 2T^{2} \) |
| 3 | \( 1 + 0.700T + 3T^{2} \) |
| 5 | \( 1 - 0.497T + 5T^{2} \) |
| 7 | \( 1 + 5.07T + 7T^{2} \) |
| 11 | \( 1 + 0.805T + 11T^{2} \) |
| 13 | \( 1 - 1.59T + 13T^{2} \) |
| 19 | \( 1 - 3.21T + 19T^{2} \) |
| 23 | \( 1 - 0.725T + 23T^{2} \) |
| 29 | \( 1 - 7.95T + 29T^{2} \) |
| 31 | \( 1 + 3.36T + 31T^{2} \) |
| 37 | \( 1 - 9.30T + 37T^{2} \) |
| 41 | \( 1 + 3.11T + 41T^{2} \) |
| 43 | \( 1 - 0.820T + 43T^{2} \) |
| 47 | \( 1 - 7.08T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 61 | \( 1 + 6.06T + 61T^{2} \) |
| 67 | \( 1 + 4.30T + 67T^{2} \) |
| 71 | \( 1 + 4.14T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 1.85T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 - 3.22T + 89T^{2} \) |
| 97 | \( 1 - 16.7T + 97T^{2} \) |
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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
−9.832003644149152658344722943745, −9.300913786445848681423475607945, −8.511803250699211631942977722340, −7.44486360828444491711708925518, −6.11132902114718985525443219438, −5.95631468673765902156978839111, −4.78464988324361169307615886004, −3.56721375150787558728405971477, −2.88911740608050364114921697264, −0.61347257187322866516083345342,
0.61347257187322866516083345342, 2.88911740608050364114921697264, 3.56721375150787558728405971477, 4.78464988324361169307615886004, 5.95631468673765902156978839111, 6.11132902114718985525443219438, 7.44486360828444491711708925518, 8.511803250699211631942977722340, 9.300913786445848681423475607945, 9.832003644149152658344722943745
