| 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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 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
−7.51004794245300332510814556698, −6.87926873158485228615711896299, −5.88690454098583158593947203225, −5.31336286791762821785831351647, −4.60338040890029151183753796142, −3.38068263771841829035874082593, −3.08094413616258763409053684256, −2.34728248104192522388135277053, −1.55572275023715000463344895741, 0,
1.55572275023715000463344895741, 2.34728248104192522388135277053, 3.08094413616258763409053684256, 3.38068263771841829035874082593, 4.60338040890029151183753796142, 5.31336286791762821785831351647, 5.88690454098583158593947203225, 6.87926873158485228615711896299, 7.51004794245300332510814556698
