| L(s) = 1 | + 3-s + 2.45·5-s + 0.714·7-s + 9-s + 1.07·11-s − 6.69·13-s + 2.45·15-s + 1.59·17-s − 5.29·19-s + 0.714·21-s − 0.914·23-s + 1.04·25-s + 27-s − 8.70·29-s − 7.73·31-s + 1.07·33-s + 1.75·35-s − 5.65·37-s − 6.69·39-s + 5.51·41-s + 6.28·43-s + 2.45·45-s − 0.0858·47-s − 6.49·49-s + 1.59·51-s − 0.447·53-s + 2.65·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.09·5-s + 0.269·7-s + 0.333·9-s + 0.325·11-s − 1.85·13-s + 0.635·15-s + 0.387·17-s − 1.21·19-s + 0.155·21-s − 0.190·23-s + 0.209·25-s + 0.192·27-s − 1.61·29-s − 1.38·31-s + 0.187·33-s + 0.296·35-s − 0.928·37-s − 1.07·39-s + 0.860·41-s + 0.958·43-s + 0.366·45-s − 0.0125·47-s − 0.927·49-s + 0.223·51-s − 0.0614·53-s + 0.357·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 2.45T + 5T^{2} \) |
| 7 | \( 1 - 0.714T + 7T^{2} \) |
| 11 | \( 1 - 1.07T + 11T^{2} \) |
| 13 | \( 1 + 6.69T + 13T^{2} \) |
| 17 | \( 1 - 1.59T + 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 + 0.914T + 23T^{2} \) |
| 29 | \( 1 + 8.70T + 29T^{2} \) |
| 31 | \( 1 + 7.73T + 31T^{2} \) |
| 37 | \( 1 + 5.65T + 37T^{2} \) |
| 41 | \( 1 - 5.51T + 41T^{2} \) |
| 43 | \( 1 - 6.28T + 43T^{2} \) |
| 47 | \( 1 + 0.0858T + 47T^{2} \) |
| 53 | \( 1 + 0.447T + 53T^{2} \) |
| 59 | \( 1 + 4.05T + 59T^{2} \) |
| 61 | \( 1 + 5.73T + 61T^{2} \) |
| 67 | \( 1 - 9.03T + 67T^{2} \) |
| 71 | \( 1 + 0.319T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 - 2.88T + 83T^{2} \) |
| 89 | \( 1 + 5.37T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.41205413530204142475211861054, −6.99227947606408764011078385951, −5.97133742817240808682239217446, −5.48850478540892393112208010586, −4.65739688310274675370282044392, −3.93548447082642817480982984620, −2.91866878372308806186805249617, −2.08369777467016710936528741599, −1.70695079712820841576472931519, 0,
1.70695079712820841576472931519, 2.08369777467016710936528741599, 2.91866878372308806186805249617, 3.93548447082642817480982984620, 4.65739688310274675370282044392, 5.48850478540892393112208010586, 5.97133742817240808682239217446, 6.99227947606408764011078385951, 7.41205413530204142475211861054
