| L(s) = 1 | + 2.90·3-s − 5-s − 2·7-s + 5.42·9-s − 0.903·11-s − 13-s − 2.90·15-s − 5.05·17-s − 0.903·19-s − 5.80·21-s − 4.14·23-s + 25-s + 7.05·27-s − 5.18·29-s + 2.14·31-s − 2.62·33-s + 2·35-s − 9.47·37-s − 2.90·39-s + 5.05·41-s − 0.147·43-s − 5.42·45-s − 4.75·47-s − 3·49-s − 14.6·51-s + 3.80·53-s + 0.903·55-s + ⋯ |
| L(s) = 1 | + 1.67·3-s − 0.447·5-s − 0.755·7-s + 1.80·9-s − 0.272·11-s − 0.277·13-s − 0.749·15-s − 1.22·17-s − 0.207·19-s − 1.26·21-s − 0.864·23-s + 0.200·25-s + 1.35·27-s − 0.962·29-s + 0.385·31-s − 0.456·33-s + 0.338·35-s − 1.55·37-s − 0.464·39-s + 0.788·41-s − 0.0225·43-s − 0.809·45-s − 0.693·47-s − 0.428·49-s − 2.05·51-s + 0.522·53-s + 0.121·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.90T + 3T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 0.903T + 11T^{2} \) |
| 17 | \( 1 + 5.05T + 17T^{2} \) |
| 19 | \( 1 + 0.903T + 19T^{2} \) |
| 23 | \( 1 + 4.14T + 23T^{2} \) |
| 29 | \( 1 + 5.18T + 29T^{2} \) |
| 31 | \( 1 - 2.14T + 31T^{2} \) |
| 37 | \( 1 + 9.47T + 37T^{2} \) |
| 41 | \( 1 - 5.05T + 41T^{2} \) |
| 43 | \( 1 + 0.147T + 43T^{2} \) |
| 47 | \( 1 + 4.75T + 47T^{2} \) |
| 53 | \( 1 - 3.80T + 53T^{2} \) |
| 59 | \( 1 + 2.70T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 + 1.85T + 71T^{2} \) |
| 73 | \( 1 + 7.37T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 8.10T + 83T^{2} \) |
| 89 | \( 1 - 8.36T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−8.165586843199934613929768331337, −7.41775642731824672714046606531, −6.88626241755345564706537674578, −5.97279870194189594886933812090, −4.74681741671350292955430272202, −3.98349895576892032263441421187, −3.35505218234393207587257982514, −2.55603868433244882877832320772, −1.81042896300726631976392781590, 0,
1.81042896300726631976392781590, 2.55603868433244882877832320772, 3.35505218234393207587257982514, 3.98349895576892032263441421187, 4.74681741671350292955430272202, 5.97279870194189594886933812090, 6.88626241755345564706537674578, 7.41775642731824672714046606531, 8.165586843199934613929768331337
