| L(s) = 1 | − 3·3-s − 10.2·5-s − 24.9·7-s + 9·9-s − 15.9·11-s − 13·13-s + 30.7·15-s − 22.8·17-s − 74.8·19-s + 74.9·21-s + 123.·23-s − 19.6·25-s − 27·27-s + 149.·29-s + 154.·31-s + 47.8·33-s + 256.·35-s + 321.·37-s + 39·39-s + 271.·41-s − 346.·43-s − 92.3·45-s + 80.7·47-s + 281.·49-s + 68.4·51-s + 38.9·53-s + 163.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.918·5-s − 1.34·7-s + 0.333·9-s − 0.436·11-s − 0.277·13-s + 0.530·15-s − 0.325·17-s − 0.903·19-s + 0.779·21-s + 1.12·23-s − 0.156·25-s − 0.192·27-s + 0.956·29-s + 0.894·31-s + 0.252·33-s + 1.23·35-s + 1.42·37-s + 0.160·39-s + 1.03·41-s − 1.23·43-s − 0.306·45-s + 0.250·47-s + 0.820·49-s + 0.188·51-s + 0.101·53-s + 0.401·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 + 10.2T + 125T^{2} \) |
| 7 | \( 1 + 24.9T + 343T^{2} \) |
| 11 | \( 1 + 15.9T + 1.33e3T^{2} \) |
| 17 | \( 1 + 22.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 74.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 123.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 149.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 154.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 321.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 271.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 346.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 80.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 38.9T + 1.48e5T^{2} \) |
| 59 | \( 1 - 47.9T + 2.05e5T^{2} \) |
| 61 | \( 1 + 320.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 580.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 82.0T + 3.57e5T^{2} \) |
| 73 | \( 1 - 547.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 580.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 99.3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 318.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.78e3T + 9.12e5T^{2} \) |
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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
−8.104642120056510177728233415277, −7.35859858436548976266177131616, −6.54839260527112439028378851516, −6.09263082315025678346337065506, −4.90444933304197063181522102519, −4.25087129426026048621599259592, −3.29013266416320735295626740741, −2.48068180281773523262630145637, −0.808113585602111689280444837781, 0,
0.808113585602111689280444837781, 2.48068180281773523262630145637, 3.29013266416320735295626740741, 4.25087129426026048621599259592, 4.90444933304197063181522102519, 6.09263082315025678346337065506, 6.54839260527112439028378851516, 7.35859858436548976266177131616, 8.104642120056510177728233415277
