| L(s) = 1 | − 1.83·3-s + 3.65·5-s + 0.00747·7-s + 0.382·9-s + 2.08·11-s + 6.39·13-s − 6.72·15-s + 5.01·17-s + 6.01·19-s − 0.0137·21-s + 3.20·23-s + 8.38·25-s + 4.81·27-s + 6.81·29-s − 4.27·31-s − 3.83·33-s + 0.0273·35-s + 3.40·37-s − 11.7·39-s + 5.36·43-s + 1.39·45-s + 7.00·47-s − 6.99·49-s − 9.22·51-s − 6.55·53-s + 7.62·55-s − 11.0·57-s + ⋯ |
| L(s) = 1 | − 1.06·3-s + 1.63·5-s + 0.00282·7-s + 0.127·9-s + 0.628·11-s + 1.77·13-s − 1.73·15-s + 1.21·17-s + 1.38·19-s − 0.00300·21-s + 0.667·23-s + 1.67·25-s + 0.926·27-s + 1.26·29-s − 0.768·31-s − 0.667·33-s + 0.00462·35-s + 0.559·37-s − 1.88·39-s + 0.817·43-s + 0.208·45-s + 1.02·47-s − 0.999·49-s − 1.29·51-s − 0.900·53-s + 1.02·55-s − 1.46·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.648370791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.648370791\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 \) |
| good | 3 | \( 1 + 1.83T + 3T^{2} \) |
| 5 | \( 1 - 3.65T + 5T^{2} \) |
| 7 | \( 1 - 0.00747T + 7T^{2} \) |
| 11 | \( 1 - 2.08T + 11T^{2} \) |
| 13 | \( 1 - 6.39T + 13T^{2} \) |
| 17 | \( 1 - 5.01T + 17T^{2} \) |
| 19 | \( 1 - 6.01T + 19T^{2} \) |
| 23 | \( 1 - 3.20T + 23T^{2} \) |
| 29 | \( 1 - 6.81T + 29T^{2} \) |
| 31 | \( 1 + 4.27T + 31T^{2} \) |
| 37 | \( 1 - 3.40T + 37T^{2} \) |
| 43 | \( 1 - 5.36T + 43T^{2} \) |
| 47 | \( 1 - 7.00T + 47T^{2} \) |
| 53 | \( 1 + 6.55T + 53T^{2} \) |
| 59 | \( 1 + 0.883T + 59T^{2} \) |
| 61 | \( 1 + 4.07T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 + 16.1T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 1.78T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
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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
−7.969897500366628226001831297202, −6.99511714810727378722517465858, −6.25449671979448700766998661583, −5.90562621334505176718400130544, −5.43600500859164487608033730410, −4.67841724114800046092449465882, −3.49020933085181273548515281572, −2.77577251071269245809016192532, −1.30121961782416132815493959402, −1.13828359205031031287000205417,
1.13828359205031031287000205417, 1.30121961782416132815493959402, 2.77577251071269245809016192532, 3.49020933085181273548515281572, 4.67841724114800046092449465882, 5.43600500859164487608033730410, 5.90562621334505176718400130544, 6.25449671979448700766998661583, 6.99511714810727378722517465858, 7.969897500366628226001831297202
