| L(s) = 1 | + 5.40·3-s − 5·5-s − 15.8·7-s + 2.26·9-s + 20.7·11-s + 40.3·13-s − 27.0·15-s + 40.6·17-s − 27.4·19-s − 85.6·21-s − 144.·23-s + 25·25-s − 133.·27-s − 29·29-s + 190.·31-s + 112.·33-s + 79.1·35-s + 65.8·37-s + 218.·39-s − 291.·41-s − 274.·43-s − 11.3·45-s + 32.1·47-s − 92.4·49-s + 219.·51-s + 407.·53-s − 103.·55-s + ⋯ |
| L(s) = 1 | + 1.04·3-s − 0.447·5-s − 0.854·7-s + 0.0837·9-s + 0.567·11-s + 0.860·13-s − 0.465·15-s + 0.580·17-s − 0.330·19-s − 0.889·21-s − 1.30·23-s + 0.200·25-s − 0.953·27-s − 0.185·29-s + 1.10·31-s + 0.590·33-s + 0.382·35-s + 0.292·37-s + 0.895·39-s − 1.10·41-s − 0.974·43-s − 0.0374·45-s + 0.0999·47-s − 0.269·49-s + 0.603·51-s + 1.05·53-s − 0.253·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 - 5.40T + 27T^{2} \) |
| 7 | \( 1 + 15.8T + 343T^{2} \) |
| 11 | \( 1 - 20.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 40.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 40.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 27.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 144.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 190.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 65.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 291.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 274.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 32.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 407.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 223.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 717.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 684.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 462.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 551.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 529.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 814.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 24.8T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.49e3T + 9.12e5T^{2} \) |
| 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
−8.816436108257124544102847005812, −8.360867974740071791021955013109, −7.52341676533148195407732181506, −6.50518415325226059004726269437, −5.79866407409173403305738799128, −4.29773992653080921008112919019, −3.54428738920380317610729779475, −2.85530550695809413871618828389, −1.54090611018359539899695571423, 0,
1.54090611018359539899695571423, 2.85530550695809413871618828389, 3.54428738920380317610729779475, 4.29773992653080921008112919019, 5.79866407409173403305738799128, 6.50518415325226059004726269437, 7.52341676533148195407732181506, 8.360867974740071791021955013109, 8.816436108257124544102847005812
