| L(s) = 1 | + 2.41·3-s + 2.82·5-s − 4.41·7-s + 2.82·9-s − 3.24·11-s + 6.82·15-s − 5.82·17-s + 1.24·19-s − 10.6·21-s − 1.24·23-s + 3.00·25-s − 0.414·27-s + 8.65·29-s − 5.65·31-s − 7.82·33-s − 12.4·35-s − 7.48·37-s − 5.82·41-s − 4.07·43-s + 8·45-s − 6·47-s + 12.4·49-s − 14.0·51-s − 2.82·53-s − 9.17·55-s + 3·57-s + 1.24·59-s + ⋯ |
| L(s) = 1 | + 1.39·3-s + 1.26·5-s − 1.66·7-s + 0.942·9-s − 0.977·11-s + 1.76·15-s − 1.41·17-s + 0.285·19-s − 2.32·21-s − 0.259·23-s + 0.600·25-s − 0.0797·27-s + 1.60·29-s − 1.01·31-s − 1.36·33-s − 2.11·35-s − 1.23·37-s − 0.910·41-s − 0.620·43-s + 1.19·45-s − 0.875·47-s + 1.78·49-s − 1.97·51-s − 0.388·53-s − 1.23·55-s + 0.397·57-s + 0.161·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2.41T + 3T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 + 4.41T + 7T^{2} \) |
| 11 | \( 1 + 3.24T + 11T^{2} \) |
| 17 | \( 1 + 5.82T + 17T^{2} \) |
| 19 | \( 1 - 1.24T + 19T^{2} \) |
| 23 | \( 1 + 1.24T + 23T^{2} \) |
| 29 | \( 1 - 8.65T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + 7.48T + 37T^{2} \) |
| 41 | \( 1 + 5.82T + 41T^{2} \) |
| 43 | \( 1 + 4.07T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 2.82T + 53T^{2} \) |
| 59 | \( 1 - 1.24T + 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 - 7.24T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 3.34T + 89T^{2} \) |
| 97 | \( 1 + 9T + 97T^{2} \) |
| 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
−8.017101446729315471401142100435, −6.88862185089228911593369873000, −6.62594493305869202975504281596, −5.72034789108625399316460283236, −4.93432526474318105943096375768, −3.76791580286838541430769164191, −3.06122981171608459847492081771, −2.49754545051375097154283043208, −1.80268290889182784593386835245, 0,
1.80268290889182784593386835245, 2.49754545051375097154283043208, 3.06122981171608459847492081771, 3.76791580286838541430769164191, 4.93432526474318105943096375768, 5.72034789108625399316460283236, 6.62594493305869202975504281596, 6.88862185089228911593369873000, 8.017101446729315471401142100435
