| L(s) = 1 | + 2.32·3-s − 1.27·5-s + 7-s + 2.41·9-s + 6.42·11-s − 2.96·15-s + 3.33·17-s − 1.32·19-s + 2.32·21-s + 0.849·23-s − 3.38·25-s − 1.36·27-s + 8.55·29-s + 1.84·31-s + 14.9·33-s − 1.27·35-s + 9.91·37-s + 2.84·41-s − 6.03·43-s − 3.07·45-s + 4.70·47-s + 49-s + 7.75·51-s − 4.57·53-s − 8.17·55-s − 3.07·57-s − 1.56·59-s + ⋯ |
| L(s) = 1 | + 1.34·3-s − 0.569·5-s + 0.377·7-s + 0.804·9-s + 1.93·11-s − 0.764·15-s + 0.808·17-s − 0.303·19-s + 0.507·21-s + 0.177·23-s − 0.676·25-s − 0.262·27-s + 1.58·29-s + 0.330·31-s + 2.60·33-s − 0.215·35-s + 1.63·37-s + 0.444·41-s − 0.920·43-s − 0.457·45-s + 0.686·47-s + 0.142·49-s + 1.08·51-s − 0.628·53-s − 1.10·55-s − 0.407·57-s − 0.203·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.025874205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.025874205\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2.32T + 3T^{2} \) |
| 5 | \( 1 + 1.27T + 5T^{2} \) |
| 11 | \( 1 - 6.42T + 11T^{2} \) |
| 17 | \( 1 - 3.33T + 17T^{2} \) |
| 19 | \( 1 + 1.32T + 19T^{2} \) |
| 23 | \( 1 - 0.849T + 23T^{2} \) |
| 29 | \( 1 - 8.55T + 29T^{2} \) |
| 31 | \( 1 - 1.84T + 31T^{2} \) |
| 37 | \( 1 - 9.91T + 37T^{2} \) |
| 41 | \( 1 - 2.84T + 41T^{2} \) |
| 43 | \( 1 + 6.03T + 43T^{2} \) |
| 47 | \( 1 - 4.70T + 47T^{2} \) |
| 53 | \( 1 + 4.57T + 53T^{2} \) |
| 59 | \( 1 + 1.56T + 59T^{2} \) |
| 61 | \( 1 + 7.50T + 61T^{2} \) |
| 67 | \( 1 - 1.34T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 6.31T + 79T^{2} \) |
| 83 | \( 1 - 5.72T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
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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
−8.032986071552843840616852021755, −7.14635471078146724265351701255, −6.48529743993944594985648333307, −5.75932495653361353152594192157, −4.50884327480952870589615025174, −4.15357343154830688299252115828, −3.40675121405984641990489520006, −2.76790720522020969243659349312, −1.75445512567135327024647050097, −0.965439237041481478953175800237,
0.965439237041481478953175800237, 1.75445512567135327024647050097, 2.76790720522020969243659349312, 3.40675121405984641990489520006, 4.15357343154830688299252115828, 4.50884327480952870589615025174, 5.75932495653361353152594192157, 6.48529743993944594985648333307, 7.14635471078146724265351701255, 8.032986071552843840616852021755
