| L(s) = 1 | − 2-s + 1.61·3-s + 4-s − 1.61·6-s + 3.23·7-s − 8-s − 0.381·9-s − 1.38·11-s + 1.61·12-s + 2.85·13-s − 3.23·14-s + 16-s + 4.47·17-s + 0.381·18-s + 4.47·19-s + 5.23·21-s + 1.38·22-s − 2.85·23-s − 1.61·24-s − 2.85·26-s − 5.47·27-s + 3.23·28-s − 9.32·29-s + 7.38·31-s − 32-s − 2.23·33-s − 4.47·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.934·3-s + 0.5·4-s − 0.660·6-s + 1.22·7-s − 0.353·8-s − 0.127·9-s − 0.416·11-s + 0.467·12-s + 0.791·13-s − 0.864·14-s + 0.250·16-s + 1.08·17-s + 0.0900·18-s + 1.02·19-s + 1.14·21-s + 0.294·22-s − 0.595·23-s − 0.330·24-s − 0.559·26-s − 1.05·27-s + 0.611·28-s − 1.73·29-s + 1.32·31-s − 0.176·32-s − 0.389·33-s − 0.766·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.055182225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.055182225\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 11 | \( 1 + 1.38T + 11T^{2} \) |
| 13 | \( 1 - 2.85T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 + 2.85T + 23T^{2} \) |
| 29 | \( 1 + 9.32T + 29T^{2} \) |
| 31 | \( 1 - 7.38T + 31T^{2} \) |
| 41 | \( 1 - 9.61T + 41T^{2} \) |
| 43 | \( 1 - 5.23T + 43T^{2} \) |
| 47 | \( 1 - 1.23T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 + 4.76T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 1.09T + 67T^{2} \) |
| 71 | \( 1 - 2.94T + 71T^{2} \) |
| 73 | \( 1 + 7.09T + 73T^{2} \) |
| 79 | \( 1 + 8.56T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 + 1.52T + 89T^{2} \) |
| 97 | \( 1 - 0.472T + 97T^{2} \) |
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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
−9.163989806725291634246632017566, −8.345416302891367089728665455621, −7.83845619818879868059911359685, −7.43754249445562827816109448725, −5.98503522449555619853665348234, −5.36115706059252588801950825631, −4.07417128609313017282858734365, −3.12247841847445430566933046554, −2.16233851146825294382014338910, −1.11681020921238536563500437236,
1.11681020921238536563500437236, 2.16233851146825294382014338910, 3.12247841847445430566933046554, 4.07417128609313017282858734365, 5.36115706059252588801950825631, 5.98503522449555619853665348234, 7.43754249445562827816109448725, 7.83845619818879868059911359685, 8.345416302891367089728665455621, 9.163989806725291634246632017566
