| L(s) = 1 | − 3-s + 4.23·5-s − 0.513·7-s + 9-s − 2.27·11-s + 2.85·13-s − 4.23·15-s + 6.71·17-s + 6.76·19-s + 0.513·21-s + 0.132·23-s + 12.9·25-s − 27-s − 2.95·29-s − 2.49·31-s + 2.27·33-s − 2.17·35-s + 4.59·37-s − 2.85·39-s + 2.66·41-s + 8.46·43-s + 4.23·45-s − 5.38·47-s − 6.73·49-s − 6.71·51-s − 4.52·53-s − 9.64·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.89·5-s − 0.194·7-s + 0.333·9-s − 0.686·11-s + 0.792·13-s − 1.09·15-s + 1.62·17-s + 1.55·19-s + 0.112·21-s + 0.0276·23-s + 2.58·25-s − 0.192·27-s − 0.549·29-s − 0.447·31-s + 0.396·33-s − 0.367·35-s + 0.754·37-s − 0.457·39-s + 0.415·41-s + 1.29·43-s + 0.631·45-s − 0.784·47-s − 0.962·49-s − 0.940·51-s − 0.621·53-s − 1.30·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.757687470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.757687470\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 503 | \( 1 + T \) |
| good | 5 | \( 1 - 4.23T + 5T^{2} \) |
| 7 | \( 1 + 0.513T + 7T^{2} \) |
| 11 | \( 1 + 2.27T + 11T^{2} \) |
| 13 | \( 1 - 2.85T + 13T^{2} \) |
| 17 | \( 1 - 6.71T + 17T^{2} \) |
| 19 | \( 1 - 6.76T + 19T^{2} \) |
| 23 | \( 1 - 0.132T + 23T^{2} \) |
| 29 | \( 1 + 2.95T + 29T^{2} \) |
| 31 | \( 1 + 2.49T + 31T^{2} \) |
| 37 | \( 1 - 4.59T + 37T^{2} \) |
| 41 | \( 1 - 2.66T + 41T^{2} \) |
| 43 | \( 1 - 8.46T + 43T^{2} \) |
| 47 | \( 1 + 5.38T + 47T^{2} \) |
| 53 | \( 1 + 4.52T + 53T^{2} \) |
| 59 | \( 1 - 3.87T + 59T^{2} \) |
| 61 | \( 1 + 7.60T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 + 7.80T + 71T^{2} \) |
| 73 | \( 1 - 7.62T + 73T^{2} \) |
| 79 | \( 1 - 7.12T + 79T^{2} \) |
| 83 | \( 1 - 4.86T + 83T^{2} \) |
| 89 | \( 1 + 3.93T + 89T^{2} \) |
| 97 | \( 1 - 5.48T + 97T^{2} \) |
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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
−7.905739486182863084203328703369, −7.36812781788711675155468815134, −6.35584581152009347345860211127, −5.84466586445594037143140798351, −5.45702468034318644182195453489, −4.79267510346958939694965836838, −3.44372084199777880765204336094, −2.77556775280488823320375557760, −1.65687810650294570286912185829, −0.987414549523421277293288386132,
0.987414549523421277293288386132, 1.65687810650294570286912185829, 2.77556775280488823320375557760, 3.44372084199777880765204336094, 4.79267510346958939694965836838, 5.45702468034318644182195453489, 5.84466586445594037143140798351, 6.35584581152009347345860211127, 7.36812781788711675155468815134, 7.905739486182863084203328703369
