| L(s) = 1 | + 2.80·2-s + 0.554·3-s + 5.85·4-s + 1.55·6-s − 3.04·7-s + 10.7·8-s − 2.69·9-s − 2.93·11-s + 3.24·12-s + 3.24·13-s − 8.54·14-s + 18.5·16-s − 2.15·17-s − 7.54·18-s − 19-s − 1.69·21-s − 8.23·22-s + 1.19·23-s + 5.98·24-s + 9.09·26-s − 3.15·27-s − 17.8·28-s − 1.77·29-s − 9.34·31-s + 30.3·32-s − 1.63·33-s − 6.04·34-s + ⋯ |
| L(s) = 1 | + 1.98·2-s + 0.320·3-s + 2.92·4-s + 0.634·6-s − 1.15·7-s + 3.81·8-s − 0.897·9-s − 0.886·11-s + 0.937·12-s + 0.900·13-s − 2.28·14-s + 4.63·16-s − 0.523·17-s − 1.77·18-s − 0.229·19-s − 0.369·21-s − 1.75·22-s + 0.249·23-s + 1.22·24-s + 1.78·26-s − 0.607·27-s − 3.37·28-s − 0.329·29-s − 1.67·31-s + 5.36·32-s − 0.283·33-s − 1.03·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.352259865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.352259865\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.80T + 2T^{2} \) |
| 3 | \( 1 - 0.554T + 3T^{2} \) |
| 7 | \( 1 + 3.04T + 7T^{2} \) |
| 11 | \( 1 + 2.93T + 11T^{2} \) |
| 13 | \( 1 - 3.24T + 13T^{2} \) |
| 17 | \( 1 + 2.15T + 17T^{2} \) |
| 23 | \( 1 - 1.19T + 23T^{2} \) |
| 29 | \( 1 + 1.77T + 29T^{2} \) |
| 31 | \( 1 + 9.34T + 31T^{2} \) |
| 37 | \( 1 + 1.15T + 37T^{2} \) |
| 41 | \( 1 - 8.57T + 41T^{2} \) |
| 43 | \( 1 - 5.27T + 43T^{2} \) |
| 47 | \( 1 - 2.35T + 47T^{2} \) |
| 53 | \( 1 - 8.82T + 53T^{2} \) |
| 59 | \( 1 + 5.70T + 59T^{2} \) |
| 61 | \( 1 + 9.96T + 61T^{2} \) |
| 67 | \( 1 - 4.98T + 67T^{2} \) |
| 71 | \( 1 - 2.70T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 - 5.66T + 79T^{2} \) |
| 83 | \( 1 + 3.00T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 - 3.24T + 97T^{2} \) |
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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
−11.03986268066041007923678197459, −10.76096453834377468072192542832, −9.275746539685960745523985903126, −7.941645861540312886436099168519, −6.93339324115327041452993030567, −6.00895571247158549336326661072, −5.42086286315542444767795729610, −4.03547890334950797372237490482, −3.21317720562659336125363041437, −2.33701947189199226304313544048,
2.33701947189199226304313544048, 3.21317720562659336125363041437, 4.03547890334950797372237490482, 5.42086286315542444767795729610, 6.00895571247158549336326661072, 6.93339324115327041452993030567, 7.941645861540312886436099168519, 9.275746539685960745523985903126, 10.76096453834377468072192542832, 11.03986268066041007923678197459
