| L(s) = 1 | − 2.77·3-s + 5-s − 4.55·7-s + 4.68·9-s + 2.90·13-s − 2.77·15-s − 7.08·17-s − 7.14·19-s + 12.6·21-s − 1.86·23-s + 25-s − 4.66·27-s − 1.01·29-s + 4.41·31-s − 4.55·35-s − 5.39·37-s − 8.03·39-s − 4.48·41-s − 6.73·43-s + 4.68·45-s + 7.85·47-s + 13.7·49-s + 19.6·51-s + 0.714·53-s + 19.8·57-s − 3.80·59-s + 2.11·61-s + ⋯ |
| L(s) = 1 | − 1.60·3-s + 0.447·5-s − 1.72·7-s + 1.56·9-s + 0.804·13-s − 0.715·15-s − 1.71·17-s − 1.63·19-s + 2.75·21-s − 0.389·23-s + 0.200·25-s − 0.898·27-s − 0.187·29-s + 0.792·31-s − 0.769·35-s − 0.886·37-s − 1.28·39-s − 0.700·41-s − 1.02·43-s + 0.698·45-s + 1.14·47-s + 1.96·49-s + 2.75·51-s + 0.0981·53-s + 2.62·57-s − 0.495·59-s + 0.270·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2990878485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2990878485\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.77T + 3T^{2} \) |
| 7 | \( 1 + 4.55T + 7T^{2} \) |
| 13 | \( 1 - 2.90T + 13T^{2} \) |
| 17 | \( 1 + 7.08T + 17T^{2} \) |
| 19 | \( 1 + 7.14T + 19T^{2} \) |
| 23 | \( 1 + 1.86T + 23T^{2} \) |
| 29 | \( 1 + 1.01T + 29T^{2} \) |
| 31 | \( 1 - 4.41T + 31T^{2} \) |
| 37 | \( 1 + 5.39T + 37T^{2} \) |
| 41 | \( 1 + 4.48T + 41T^{2} \) |
| 43 | \( 1 + 6.73T + 43T^{2} \) |
| 47 | \( 1 - 7.85T + 47T^{2} \) |
| 53 | \( 1 - 0.714T + 53T^{2} \) |
| 59 | \( 1 + 3.80T + 59T^{2} \) |
| 61 | \( 1 - 2.11T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 4.21T + 71T^{2} \) |
| 73 | \( 1 - 1.36T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−8.506656640638402934495609312843, −6.90683671741945592079892237169, −6.74870158129097709822427232928, −6.09272822685841769304971132712, −5.69676990701195859990463477912, −4.56836563628216679659776758718, −4.00670807154052483719442433333, −2.86331553251046590601795291746, −1.74938886623653272021715672949, −0.31524666438443033736905871891,
0.31524666438443033736905871891, 1.74938886623653272021715672949, 2.86331553251046590601795291746, 4.00670807154052483719442433333, 4.56836563628216679659776758718, 5.69676990701195859990463477912, 6.09272822685841769304971132712, 6.74870158129097709822427232928, 6.90683671741945592079892237169, 8.506656640638402934495609312843
