| L(s) = 1 | + 2.03·2-s + 2.12·4-s − 0.504·7-s + 0.248·8-s − 3.43·11-s + 4.93·13-s − 1.02·14-s − 3.74·16-s − 1.17·17-s − 5.93·19-s − 6.98·22-s − 2.97·23-s + 10.0·26-s − 1.07·28-s + 5.01·29-s + 3.56·31-s − 8.09·32-s − 2.39·34-s − 1.50·37-s − 12.0·38-s + 8.78·41-s − 10.2·43-s − 7.29·44-s − 6.04·46-s + 2.85·47-s − 6.74·49-s + 10.4·52-s + ⋯ |
| L(s) = 1 | + 1.43·2-s + 1.06·4-s − 0.190·7-s + 0.0878·8-s − 1.03·11-s + 1.36·13-s − 0.273·14-s − 0.935·16-s − 0.285·17-s − 1.36·19-s − 1.48·22-s − 0.620·23-s + 1.96·26-s − 0.202·28-s + 0.931·29-s + 0.639·31-s − 1.43·32-s − 0.409·34-s − 0.246·37-s − 1.95·38-s + 1.37·41-s − 1.56·43-s − 1.10·44-s − 0.891·46-s + 0.416·47-s − 0.963·49-s + 1.45·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.03T + 2T^{2} \) |
| 7 | \( 1 + 0.504T + 7T^{2} \) |
| 11 | \( 1 + 3.43T + 11T^{2} \) |
| 13 | \( 1 - 4.93T + 13T^{2} \) |
| 17 | \( 1 + 1.17T + 17T^{2} \) |
| 19 | \( 1 + 5.93T + 19T^{2} \) |
| 23 | \( 1 + 2.97T + 23T^{2} \) |
| 29 | \( 1 - 5.01T + 29T^{2} \) |
| 31 | \( 1 - 3.56T + 31T^{2} \) |
| 37 | \( 1 + 1.50T + 37T^{2} \) |
| 41 | \( 1 - 8.78T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 2.85T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 - 9.50T + 59T^{2} \) |
| 61 | \( 1 + 8.48T + 61T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 + 0.356T + 83T^{2} \) |
| 89 | \( 1 + 6.51T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−7.70205470612534870226042589258, −6.65597289637244136185386259010, −6.17383714854858144713872484307, −5.67414042098499942157774975274, −4.61877880836726131125738602018, −4.32116582628114715068735023681, −3.30254702518341889880034646439, −2.73729239351632215773182247764, −1.69625251481782089581019609666, 0,
1.69625251481782089581019609666, 2.73729239351632215773182247764, 3.30254702518341889880034646439, 4.32116582628114715068735023681, 4.61877880836726131125738602018, 5.67414042098499942157774975274, 6.17383714854858144713872484307, 6.65597289637244136185386259010, 7.70205470612534870226042589258
