| L(s) = 1 | − 1.13·3-s + 5-s − 7-s − 1.70·9-s − 3.23·11-s + 13-s − 1.13·15-s + 2.98·17-s + 1.93·19-s + 1.13·21-s − 2.17·23-s + 25-s + 5.35·27-s + 3.58·29-s + 7.29·31-s + 3.68·33-s − 35-s + 2.37·37-s − 1.13·39-s + 1.76·41-s − 5.51·43-s − 1.70·45-s − 7.53·47-s + 49-s − 3.39·51-s − 3.40·53-s − 3.23·55-s + ⋯ |
| L(s) = 1 | − 0.657·3-s + 0.447·5-s − 0.377·7-s − 0.567·9-s − 0.975·11-s + 0.277·13-s − 0.293·15-s + 0.722·17-s + 0.445·19-s + 0.248·21-s − 0.452·23-s + 0.200·25-s + 1.03·27-s + 0.666·29-s + 1.31·31-s + 0.641·33-s − 0.169·35-s + 0.390·37-s − 0.182·39-s + 0.276·41-s − 0.840·43-s − 0.253·45-s − 1.09·47-s + 0.142·49-s − 0.475·51-s − 0.468·53-s − 0.436·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 1.13T + 3T^{2} \) |
| 11 | \( 1 + 3.23T + 11T^{2} \) |
| 17 | \( 1 - 2.98T + 17T^{2} \) |
| 19 | \( 1 - 1.93T + 19T^{2} \) |
| 23 | \( 1 + 2.17T + 23T^{2} \) |
| 29 | \( 1 - 3.58T + 29T^{2} \) |
| 31 | \( 1 - 7.29T + 31T^{2} \) |
| 37 | \( 1 - 2.37T + 37T^{2} \) |
| 41 | \( 1 - 1.76T + 41T^{2} \) |
| 43 | \( 1 + 5.51T + 43T^{2} \) |
| 47 | \( 1 + 7.53T + 47T^{2} \) |
| 53 | \( 1 + 3.40T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 5.68T + 61T^{2} \) |
| 67 | \( 1 - 2.04T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 7.09T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 - 2.61T + 83T^{2} \) |
| 89 | \( 1 - 3.92T + 89T^{2} \) |
| 97 | \( 1 + 1.34T + 97T^{2} \) |
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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
−8.152493944538901833233999407831, −7.47234445650426276321879609453, −6.36266633574154458096594057647, −6.03588414994209172792264608302, −5.21834574888762375507383875854, −4.58600274389060938481661853020, −3.23211690529046058739972414562, −2.68133330691268926924509747122, −1.29273365354815276872462150311, 0,
1.29273365354815276872462150311, 2.68133330691268926924509747122, 3.23211690529046058739972414562, 4.58600274389060938481661853020, 5.21834574888762375507383875854, 6.03588414994209172792264608302, 6.36266633574154458096594057647, 7.47234445650426276321879609453, 8.152493944538901833233999407831
