| L(s) = 1 | + 0.369·2-s − 1.38·3-s − 1.86·4-s − 1.75·5-s − 0.512·6-s + 1.96·7-s − 1.42·8-s − 1.07·9-s − 0.650·10-s + 1.73·11-s + 2.58·12-s − 6.13·13-s + 0.727·14-s + 2.44·15-s + 3.19·16-s − 17-s − 0.396·18-s − 0.392·19-s + 3.27·20-s − 2.73·21-s + 0.639·22-s − 0.582·23-s + 1.98·24-s − 1.90·25-s − 2.26·26-s + 5.65·27-s − 3.67·28-s + ⋯ |
| L(s) = 1 | + 0.261·2-s − 0.801·3-s − 0.931·4-s − 0.786·5-s − 0.209·6-s + 0.744·7-s − 0.504·8-s − 0.357·9-s − 0.205·10-s + 0.521·11-s + 0.746·12-s − 1.70·13-s + 0.194·14-s + 0.630·15-s + 0.799·16-s − 0.242·17-s − 0.0934·18-s − 0.0899·19-s + 0.733·20-s − 0.596·21-s + 0.136·22-s − 0.121·23-s + 0.404·24-s − 0.380·25-s − 0.444·26-s + 1.08·27-s − 0.693·28-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
| \(L(1)\) |
\(\approx\) |
\(0.6953970730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6953970730\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.369T + 2T^{2} \) |
| 3 | \( 1 + 1.38T + 3T^{2} \) |
| 5 | \( 1 + 1.75T + 5T^{2} \) |
| 7 | \( 1 - 1.96T + 7T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 + 6.13T + 13T^{2} \) |
| 19 | \( 1 + 0.392T + 19T^{2} \) |
| 23 | \( 1 + 0.582T + 23T^{2} \) |
| 29 | \( 1 - 1.66T + 29T^{2} \) |
| 31 | \( 1 - 4.83T + 31T^{2} \) |
| 37 | \( 1 - 9.19T + 37T^{2} \) |
| 41 | \( 1 - 4.79T + 41T^{2} \) |
| 43 | \( 1 - 7.51T + 43T^{2} \) |
| 47 | \( 1 - 8.86T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 61 | \( 1 - 4.42T + 61T^{2} \) |
| 67 | \( 1 - 9.45T + 67T^{2} \) |
| 71 | \( 1 - 5.81T + 71T^{2} \) |
| 73 | \( 1 - 5.92T + 73T^{2} \) |
| 79 | \( 1 + 16.8T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 6.83T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−9.927683768295343866351071524918, −9.195885841105723318240334383420, −8.183607222721860644326307015732, −7.61934131242693123915841802994, −6.41023218381242385730600119016, −5.44892952842510114643155250125, −4.67022548326547037744185788489, −4.11673565233116912681264584094, −2.65558655285734000026616861611, −0.63908113984329678410535248502,
0.63908113984329678410535248502, 2.65558655285734000026616861611, 4.11673565233116912681264584094, 4.67022548326547037744185788489, 5.44892952842510114643155250125, 6.41023218381242385730600119016, 7.61934131242693123915841802994, 8.183607222721860644326307015732, 9.195885841105723318240334383420, 9.927683768295343866351071524918
