| L(s) = 1 | + 2.76·2-s − 1.76·3-s + 5.62·4-s + 2.62·5-s − 4.86·6-s + 10.0·8-s + 0.103·9-s + 7.25·10-s − 9.91·12-s + 2.38·13-s − 4.62·15-s + 16.4·16-s + 2.38·17-s + 0.284·18-s + 1.72·19-s + 14.7·20-s − 0.626·23-s − 17.6·24-s + 1.89·25-s + 6.59·26-s + 5.10·27-s + 1.72·29-s − 12.7·30-s + 2.23·31-s + 25.2·32-s + 6.59·34-s + 0.579·36-s + ⋯ |
| L(s) = 1 | + 1.95·2-s − 1.01·3-s + 2.81·4-s + 1.17·5-s − 1.98·6-s + 3.54·8-s + 0.0343·9-s + 2.29·10-s − 2.86·12-s + 0.662·13-s − 1.19·15-s + 4.10·16-s + 0.579·17-s + 0.0670·18-s + 0.396·19-s + 3.30·20-s − 0.130·23-s − 3.60·24-s + 0.379·25-s + 1.29·26-s + 0.982·27-s + 0.321·29-s − 2.33·30-s + 0.402·31-s + 4.46·32-s + 1.13·34-s + 0.0966·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.254501287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.254501287\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.76T + 2T^{2} \) |
| 3 | \( 1 + 1.76T + 3T^{2} \) |
| 5 | \( 1 - 2.62T + 5T^{2} \) |
| 13 | \( 1 - 2.38T + 13T^{2} \) |
| 17 | \( 1 - 2.38T + 17T^{2} \) |
| 19 | \( 1 - 1.72T + 19T^{2} \) |
| 23 | \( 1 + 0.626T + 23T^{2} \) |
| 29 | \( 1 - 1.72T + 29T^{2} \) |
| 31 | \( 1 - 2.23T + 31T^{2} \) |
| 37 | \( 1 + 6.89T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 7.25T + 43T^{2} \) |
| 47 | \( 1 + 6.38T + 47T^{2} \) |
| 53 | \( 1 + 9.25T + 53T^{2} \) |
| 59 | \( 1 - 1.76T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 6.42T + 67T^{2} \) |
| 71 | \( 1 - 8.08T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 - 8.35T + 97T^{2} \) |
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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
−7.70300675767299023842359123431, −6.78565240736717292824671070276, −6.28655911480793790405342217900, −5.86066392320886294663259073291, −5.18746775543949996433409569581, −4.85049674516802000852661351592, −3.71409577931843896424576871619, −3.04098276273533978859052543289, −2.06873546307228494018704054338, −1.22990220945194251310557139375,
1.22990220945194251310557139375, 2.06873546307228494018704054338, 3.04098276273533978859052543289, 3.71409577931843896424576871619, 4.85049674516802000852661351592, 5.18746775543949996433409569581, 5.86066392320886294663259073291, 6.28655911480793790405342217900, 6.78565240736717292824671070276, 7.70300675767299023842359123431
