| L(s) = 1 | − 0.347·3-s − 1.66·5-s + 3.92·7-s − 2.87·9-s + 6.50·11-s − 3.17·13-s + 0.579·15-s + 3.56·17-s + 8.45·19-s − 1.36·21-s − 4.47·23-s − 2.22·25-s + 2.04·27-s − 9.13·29-s − 4.27·31-s − 2.26·33-s − 6.53·35-s + 0.328·37-s + 1.10·39-s + 8.75·41-s − 4.27·43-s + 4.79·45-s + 5.78·47-s + 8.37·49-s − 1.24·51-s + 13.1·53-s − 10.8·55-s + ⋯ |
| L(s) = 1 | − 0.200·3-s − 0.745·5-s + 1.48·7-s − 0.959·9-s + 1.96·11-s − 0.879·13-s + 0.149·15-s + 0.864·17-s + 1.93·19-s − 0.297·21-s − 0.932·23-s − 0.444·25-s + 0.393·27-s − 1.69·29-s − 0.766·31-s − 0.393·33-s − 1.10·35-s + 0.0540·37-s + 0.176·39-s + 1.36·41-s − 0.651·43-s + 0.715·45-s + 0.843·47-s + 1.19·49-s − 0.173·51-s + 1.81·53-s − 1.46·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.737948729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.737948729\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 0.347T + 3T^{2} \) |
| 5 | \( 1 + 1.66T + 5T^{2} \) |
| 7 | \( 1 - 3.92T + 7T^{2} \) |
| 11 | \( 1 - 6.50T + 11T^{2} \) |
| 13 | \( 1 + 3.17T + 13T^{2} \) |
| 17 | \( 1 - 3.56T + 17T^{2} \) |
| 19 | \( 1 - 8.45T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 + 9.13T + 29T^{2} \) |
| 31 | \( 1 + 4.27T + 31T^{2} \) |
| 37 | \( 1 - 0.328T + 37T^{2} \) |
| 41 | \( 1 - 8.75T + 41T^{2} \) |
| 43 | \( 1 + 4.27T + 43T^{2} \) |
| 47 | \( 1 - 5.78T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 - 6.76T + 59T^{2} \) |
| 61 | \( 1 + 9.71T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 - 4.45T + 79T^{2} \) |
| 83 | \( 1 + 7.31T + 83T^{2} \) |
| 89 | \( 1 - 0.686T + 89T^{2} \) |
| 97 | \( 1 + 7.21T + 97T^{2} \) |
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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
−9.232884485516723771672809150131, −8.196937567297998918410270446710, −7.67480847591047737130249520644, −7.02660269666821610224505383393, −5.69206628457673461567902924743, −5.30033907243452587292247974785, −4.10846343396052631935289074380, −3.55563853231446834438843515066, −2.07333434711247556184657836428, −0.934239250573243268208994982732,
0.934239250573243268208994982732, 2.07333434711247556184657836428, 3.55563853231446834438843515066, 4.10846343396052631935289074380, 5.30033907243452587292247974785, 5.69206628457673461567902924743, 7.02660269666821610224505383393, 7.67480847591047737130249520644, 8.196937567297998918410270446710, 9.232884485516723771672809150131
