| L(s) = 1 | − 3-s − 0.425·5-s − 4.01·7-s + 9-s + 2.74·11-s + 2.75·13-s + 0.425·15-s − 0.750·17-s − 4.08·19-s + 4.01·21-s + 23-s − 4.81·25-s − 27-s − 29-s + 3.50·31-s − 2.74·33-s + 1.70·35-s − 5.25·37-s − 2.75·39-s + 11.5·41-s + 7.71·43-s − 0.425·45-s + 3.88·47-s + 9.14·49-s + 0.750·51-s − 0.561·53-s − 1.16·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.190·5-s − 1.51·7-s + 0.333·9-s + 0.828·11-s + 0.762·13-s + 0.109·15-s − 0.181·17-s − 0.937·19-s + 0.876·21-s + 0.208·23-s − 0.963·25-s − 0.192·27-s − 0.185·29-s + 0.629·31-s − 0.478·33-s + 0.289·35-s − 0.863·37-s − 0.440·39-s + 1.79·41-s + 1.17·43-s − 0.0634·45-s + 0.567·47-s + 1.30·49-s + 0.105·51-s − 0.0771·53-s − 0.157·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 0.425T + 5T^{2} \) |
| 7 | \( 1 + 4.01T + 7T^{2} \) |
| 11 | \( 1 - 2.74T + 11T^{2} \) |
| 13 | \( 1 - 2.75T + 13T^{2} \) |
| 17 | \( 1 + 0.750T + 17T^{2} \) |
| 19 | \( 1 + 4.08T + 19T^{2} \) |
| 31 | \( 1 - 3.50T + 31T^{2} \) |
| 37 | \( 1 + 5.25T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 - 7.71T + 43T^{2} \) |
| 47 | \( 1 - 3.88T + 47T^{2} \) |
| 53 | \( 1 + 0.561T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 8.17T + 73T^{2} \) |
| 79 | \( 1 + 1.31T + 79T^{2} \) |
| 83 | \( 1 - 8.92T + 83T^{2} \) |
| 89 | \( 1 - 2.90T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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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
−7.33156965625842879028396504229, −6.59563785702239970498900260745, −6.17213811706438036885292211333, −5.70818721908347095593197405392, −4.47555183968612881126617922690, −3.94666691804444338041445608259, −3.25918022167152415188277256580, −2.24369331993859501726431566492, −1.04605010374714189542577855076, 0,
1.04605010374714189542577855076, 2.24369331993859501726431566492, 3.25918022167152415188277256580, 3.94666691804444338041445608259, 4.47555183968612881126617922690, 5.70818721908347095593197405392, 6.17213811706438036885292211333, 6.59563785702239970498900260745, 7.33156965625842879028396504229
