L(s) = 1 | − 4.89·3-s − 5·5-s − 2.50·7-s − 3.03·9-s + 33.7·11-s + 10.0·13-s + 24.4·15-s + 1.45·17-s − 76.4·19-s + 12.2·21-s − 96.3·23-s + 25·25-s + 147.·27-s − 69.9·29-s + 99.0·31-s − 165.·33-s + 12.5·35-s + 285.·37-s − 49.2·39-s + 423.·41-s + 123.·43-s + 15.1·45-s − 205.·47-s − 336.·49-s − 7.11·51-s + 572.·53-s − 168.·55-s + ⋯ |
L(s) = 1 | − 0.942·3-s − 0.447·5-s − 0.135·7-s − 0.112·9-s + 0.926·11-s + 0.214·13-s + 0.421·15-s + 0.0207·17-s − 0.923·19-s + 0.127·21-s − 0.873·23-s + 0.200·25-s + 1.04·27-s − 0.447·29-s + 0.573·31-s − 0.872·33-s + 0.0604·35-s + 1.27·37-s − 0.202·39-s + 1.61·41-s + 0.437·43-s + 0.0502·45-s − 0.637·47-s − 0.981·49-s − 0.0195·51-s + 1.48·53-s − 0.414·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 5T \) |
good | 3 | \( 1 + 4.89T + 27T^{2} \) |
| 7 | \( 1 + 2.50T + 343T^{2} \) |
| 11 | \( 1 - 33.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 10.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 1.45T + 4.91e3T^{2} \) |
| 19 | \( 1 + 76.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 96.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 69.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 99.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 285.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 423.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 123.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 205.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 572.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 338.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 706.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 478.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 794.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 284.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 396.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 119.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 808.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.14e3T + 9.12e5T^{2} \) |
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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
−8.849672014486117928341001298395, −8.121731917094543281296063776892, −7.08710979953100033738738857431, −6.24654772078307060827922156124, −5.74844375577312449359911996777, −4.52427383736242079286818420814, −3.87982438826429701426414407452, −2.54983140707504792332898882529, −1.08950240274683465761509538566, 0,
1.08950240274683465761509538566, 2.54983140707504792332898882529, 3.87982438826429701426414407452, 4.52427383736242079286818420814, 5.74844375577312449359911996777, 6.24654772078307060827922156124, 7.08710979953100033738738857431, 8.121731917094543281296063776892, 8.849672014486117928341001298395