| L(s) = 1 | − 3-s − 2·4-s − 5-s − 1.73·7-s + 9-s + 2·12-s + 15-s + 4·16-s + 3.46·17-s + 5.19·19-s + 2·20-s + 1.73·21-s + 6·23-s + 25-s − 27-s + 3.46·28-s − 6.92·29-s + 31-s + 1.73·35-s − 2·36-s − 5·37-s − 3.46·41-s − 10.3·43-s − 45-s − 12·47-s − 4·48-s − 4·49-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s − 0.654·7-s + 0.333·9-s + 0.577·12-s + 0.258·15-s + 16-s + 0.840·17-s + 1.19·19-s + 0.447·20-s + 0.377·21-s + 1.25·23-s + 0.200·25-s − 0.192·27-s + 0.654·28-s − 1.28·29-s + 0.179·31-s + 0.292·35-s − 0.333·36-s − 0.821·37-s − 0.541·41-s − 1.58·43-s − 0.149·45-s − 1.75·47-s − 0.577·48-s − 0.571·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 5.19T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 5T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 1.73T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 - 6.92T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 13T + 97T^{2} \) |
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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
−8.967386097377301713078020815996, −8.096375974871344841906660400344, −7.29921933322431972523395961305, −6.48616862248567558759622718411, −5.30679273967680233329666995509, −5.04796426311487769793515776801, −3.72760061035960114855753063621, −3.23018612043223906164273475939, −1.25475203166947893171343135556, 0,
1.25475203166947893171343135556, 3.23018612043223906164273475939, 3.72760061035960114855753063621, 5.04796426311487769793515776801, 5.30679273967680233329666995509, 6.48616862248567558759622718411, 7.29921933322431972523395961305, 8.096375974871344841906660400344, 8.967386097377301713078020815996
