| L(s) = 1 | − 2.63·2-s + 3-s + 4.92·4-s − 2.63·6-s + 4.16·7-s − 7.69·8-s + 9-s + 4.92·12-s − 5.26·13-s − 10.9·14-s + 10.4·16-s − 4.23·17-s − 2.63·18-s − 2.50·19-s + 4.16·21-s − 5.48·23-s − 7.69·24-s + 13.8·26-s + 27-s + 20.5·28-s + 5.26·29-s + 10.1·31-s − 11.9·32-s + 11.1·34-s + 4.92·36-s + 4.48·37-s + 6.58·38-s + ⋯ |
| L(s) = 1 | − 1.86·2-s + 0.577·3-s + 2.46·4-s − 1.07·6-s + 1.57·7-s − 2.72·8-s + 0.333·9-s + 1.42·12-s − 1.45·13-s − 2.93·14-s + 2.60·16-s − 1.02·17-s − 0.620·18-s − 0.574·19-s + 0.909·21-s − 1.14·23-s − 1.57·24-s + 2.71·26-s + 0.192·27-s + 3.87·28-s + 0.977·29-s + 1.81·31-s − 2.12·32-s + 1.91·34-s + 0.820·36-s + 0.737·37-s + 1.06·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.078533358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.078533358\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 7 | \( 1 - 4.16T + 7T^{2} \) |
| 13 | \( 1 + 5.26T + 13T^{2} \) |
| 17 | \( 1 + 4.23T + 17T^{2} \) |
| 19 | \( 1 + 2.50T + 19T^{2} \) |
| 23 | \( 1 + 5.48T + 23T^{2} \) |
| 29 | \( 1 - 5.26T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 - 4.48T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 - 6.66T + 43T^{2} \) |
| 47 | \( 1 + 4.21T + 47T^{2} \) |
| 53 | \( 1 - 3.63T + 53T^{2} \) |
| 59 | \( 1 + 6.58T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 4.48T + 67T^{2} \) |
| 71 | \( 1 - 1.26T + 71T^{2} \) |
| 73 | \( 1 + 4.55T + 73T^{2} \) |
| 79 | \( 1 + 6.86T + 79T^{2} \) |
| 83 | \( 1 - 4.87T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−7.88848208800851134560978491370, −7.51199089959777907907062745207, −6.69308901258829304360943907875, −6.06281807497571819464475272602, −4.77320020225967937903713512082, −4.39752228812918922608429926664, −2.86157753823358127692646713741, −2.24291726086611571603406539338, −1.73337245111448785192606384295, −0.63861731197544197318058132607,
0.63861731197544197318058132607, 1.73337245111448785192606384295, 2.24291726086611571603406539338, 2.86157753823358127692646713741, 4.39752228812918922608429926664, 4.77320020225967937903713512082, 6.06281807497571819464475272602, 6.69308901258829304360943907875, 7.51199089959777907907062745207, 7.88848208800851134560978491370
