| L(s) = 1 | + 3·3-s + 22.2·7-s + 9·9-s + 1.79·11-s − 58.2·13-s − 18.9·17-s − 104.·19-s + 66.6·21-s − 49.6·23-s + 27·27-s − 293.·29-s − 64.4·31-s + 5.37·33-s + 19.8·37-s − 174.·39-s − 165.·41-s − 247.·43-s + 384.·47-s + 150.·49-s − 56.9·51-s − 463.·53-s − 314.·57-s + 73.7·59-s − 137.·61-s + 199.·63-s + 173.·67-s − 148.·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.19·7-s + 0.333·9-s + 0.0490·11-s − 1.24·13-s − 0.270·17-s − 1.26·19-s + 0.692·21-s − 0.449·23-s + 0.192·27-s − 1.87·29-s − 0.373·31-s + 0.0283·33-s + 0.0883·37-s − 0.716·39-s − 0.630·41-s − 0.877·43-s + 1.19·47-s + 0.438·49-s − 0.156·51-s − 1.20·53-s − 0.730·57-s + 0.162·59-s − 0.288·61-s + 0.399·63-s + 0.317·67-s − 0.259·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 22.2T + 343T^{2} \) |
| 11 | \( 1 - 1.79T + 1.33e3T^{2} \) |
| 13 | \( 1 + 58.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 18.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 104.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 49.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 293.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 64.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 19.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 165.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 247.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 384.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 463.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 73.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 137.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 173.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 594.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 320.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 770.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 173.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 384.T + 9.12e5T^{2} \) |
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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
−8.932228356681616004460172006328, −8.073413254060382002252391792765, −7.54485599088918513206111052289, −6.60614180145211059896373100983, −5.38574824784766810406018975284, −4.61724493613106626815371744599, −3.74636879365726534589709906161, −2.35605267086697107036003493447, −1.72066153632002513608749342338, 0,
1.72066153632002513608749342338, 2.35605267086697107036003493447, 3.74636879365726534589709906161, 4.61724493613106626815371744599, 5.38574824784766810406018975284, 6.60614180145211059896373100983, 7.54485599088918513206111052289, 8.073413254060382002252391792765, 8.932228356681616004460172006328
