| L(s) = 1 | − 2.21·3-s − 3.59·5-s + 1.90·9-s − 2.55·11-s + 4.93·13-s + 7.96·15-s − 2.68·17-s − 4.72·19-s − 1.49·23-s + 7.93·25-s + 2.41·27-s + 2.43·29-s − 3.19·31-s + 5.67·33-s + 2.90·37-s − 10.9·39-s + 41-s − 7.62·43-s − 6.86·45-s − 5.15·47-s + 5.95·51-s − 11.1·53-s + 9.20·55-s + 10.4·57-s − 1.49·59-s − 1.06·61-s − 17.7·65-s + ⋯ |
| L(s) = 1 | − 1.27·3-s − 1.60·5-s + 0.636·9-s − 0.771·11-s + 1.36·13-s + 2.05·15-s − 0.651·17-s − 1.08·19-s − 0.311·23-s + 1.58·25-s + 0.465·27-s + 0.451·29-s − 0.573·31-s + 0.987·33-s + 0.478·37-s − 1.75·39-s + 0.156·41-s − 1.16·43-s − 1.02·45-s − 0.751·47-s + 0.833·51-s − 1.52·53-s + 1.24·55-s + 1.38·57-s − 0.194·59-s − 0.136·61-s − 2.20·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1962889581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1962889581\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 2.21T + 3T^{2} \) |
| 5 | \( 1 + 3.59T + 5T^{2} \) |
| 11 | \( 1 + 2.55T + 11T^{2} \) |
| 13 | \( 1 - 4.93T + 13T^{2} \) |
| 17 | \( 1 + 2.68T + 17T^{2} \) |
| 19 | \( 1 + 4.72T + 19T^{2} \) |
| 23 | \( 1 + 1.49T + 23T^{2} \) |
| 29 | \( 1 - 2.43T + 29T^{2} \) |
| 31 | \( 1 + 3.19T + 31T^{2} \) |
| 37 | \( 1 - 2.90T + 37T^{2} \) |
| 43 | \( 1 + 7.62T + 43T^{2} \) |
| 47 | \( 1 + 5.15T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 1.49T + 59T^{2} \) |
| 61 | \( 1 + 1.06T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 + 8.28T + 73T^{2} \) |
| 79 | \( 1 + 4.28T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 9.36T + 89T^{2} \) |
| 97 | \( 1 - 3.37T + 97T^{2} \) |
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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
−7.88615856827936687100144929256, −7.01564456998242603824418094964, −6.41788510392655452298096461843, −5.84864063435797625576493704350, −4.91977417285033395412179726190, −4.39802973805081363137438000904, −3.71194421171989158426874434892, −2.84378768197958473543549434162, −1.45194768314063447941970447733, −0.24327616662904040255591992902,
0.24327616662904040255591992902, 1.45194768314063447941970447733, 2.84378768197958473543549434162, 3.71194421171989158426874434892, 4.39802973805081363137438000904, 4.91977417285033395412179726190, 5.84864063435797625576493704350, 6.41788510392655452298096461843, 7.01564456998242603824418094964, 7.88615856827936687100144929256
