| L(s) = 1 | − 2.77·3-s + 1.84·5-s + 7·7-s − 19.3·9-s + 11·11-s + 24.6·13-s − 5.11·15-s + 17.8·17-s − 32.1·19-s − 19.4·21-s − 14.1·23-s − 121.·25-s + 128.·27-s − 41.5·29-s − 175.·31-s − 30.5·33-s + 12.9·35-s + 292.·37-s − 68.3·39-s + 154.·41-s + 277.·43-s − 35.6·45-s + 52.1·47-s + 49·49-s − 49.5·51-s + 82.3·53-s + 20.2·55-s + ⋯ |
| L(s) = 1 | − 0.533·3-s + 0.164·5-s + 0.377·7-s − 0.714·9-s + 0.301·11-s + 0.525·13-s − 0.0880·15-s + 0.255·17-s − 0.388·19-s − 0.201·21-s − 0.128·23-s − 0.972·25-s + 0.915·27-s − 0.266·29-s − 1.01·31-s − 0.160·33-s + 0.0623·35-s + 1.30·37-s − 0.280·39-s + 0.587·41-s + 0.982·43-s − 0.117·45-s + 0.161·47-s + 0.142·49-s − 0.136·51-s + 0.213·53-s + 0.0497·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 + 2.77T + 27T^{2} \) |
| 5 | \( 1 - 1.84T + 125T^{2} \) |
| 13 | \( 1 - 24.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 17.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 32.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 14.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 41.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 175.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 292.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 154.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 277.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 52.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 82.3T + 1.48e5T^{2} \) |
| 59 | \( 1 + 712.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 647.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 260.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 369.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.14e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 488.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 548.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 105.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.36e3T + 9.12e5T^{2} \) |
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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
−8.980602529817914538812331368092, −8.111062501730518771677325312234, −7.32914971355681737678481346131, −6.08191703891784493496339746517, −5.82315993035821251812876127539, −4.68583120707649640975149808911, −3.73679660251847105435503951111, −2.51842369260886060443881656025, −1.30457310507710787551504086594, 0,
1.30457310507710787551504086594, 2.51842369260886060443881656025, 3.73679660251847105435503951111, 4.68583120707649640975149808911, 5.82315993035821251812876127539, 6.08191703891784493496339746517, 7.32914971355681737678481346131, 8.111062501730518771677325312234, 8.980602529817914538812331368092
