| L(s) = 1 | − 1.37·2-s − 2.46·3-s − 0.0981·4-s + 3.39·6-s + 3.48·7-s + 2.89·8-s + 3.06·9-s + 3.66·11-s + 0.241·12-s − 4.41·13-s − 4.81·14-s − 3.79·16-s + 7.90·17-s − 4.22·18-s − 4.04·19-s − 8.59·21-s − 5.05·22-s − 0.625·23-s − 7.12·24-s + 6.09·26-s − 0.160·27-s − 0.342·28-s + 10.4·29-s − 4.37·31-s − 0.554·32-s − 9.03·33-s − 10.9·34-s + ⋯ |
| L(s) = 1 | − 0.975·2-s − 1.42·3-s − 0.0490·4-s + 1.38·6-s + 1.31·7-s + 1.02·8-s + 1.02·9-s + 1.10·11-s + 0.0697·12-s − 1.22·13-s − 1.28·14-s − 0.948·16-s + 1.91·17-s − 0.996·18-s − 0.928·19-s − 1.87·21-s − 1.07·22-s − 0.130·23-s − 1.45·24-s + 1.19·26-s − 0.0308·27-s − 0.0647·28-s + 1.93·29-s − 0.785·31-s − 0.0980·32-s − 1.57·33-s − 1.87·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8565599650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8565599650\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 2 | \( 1 + 1.37T + 2T^{2} \) |
| 3 | \( 1 + 2.46T + 3T^{2} \) |
| 7 | \( 1 - 3.48T + 7T^{2} \) |
| 11 | \( 1 - 3.66T + 11T^{2} \) |
| 13 | \( 1 + 4.41T + 13T^{2} \) |
| 17 | \( 1 - 7.90T + 17T^{2} \) |
| 19 | \( 1 + 4.04T + 19T^{2} \) |
| 23 | \( 1 + 0.625T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 + 4.37T + 31T^{2} \) |
| 37 | \( 1 - 3.01T + 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 + 0.164T + 43T^{2} \) |
| 47 | \( 1 - 0.235T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 7.18T + 59T^{2} \) |
| 61 | \( 1 - 7.92T + 61T^{2} \) |
| 67 | \( 1 + 0.478T + 67T^{2} \) |
| 71 | \( 1 - 2.65T + 71T^{2} \) |
| 73 | \( 1 + 9.29T + 73T^{2} \) |
| 79 | \( 1 - 6.33T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 + 9.55T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−8.052920287226411025381601518304, −7.41445613969885487435936421338, −6.78136404695590613160172746124, −5.84832745766458635747258003562, −5.19784308722905669657200159282, −4.60326018861115323510583862993, −3.98781778358706778928104570217, −2.35620003718531785085390191258, −1.24605703898526277378222745727, −0.73767530440009910724629267095,
0.73767530440009910724629267095, 1.24605703898526277378222745727, 2.35620003718531785085390191258, 3.98781778358706778928104570217, 4.60326018861115323510583862993, 5.19784308722905669657200159282, 5.84832745766458635747258003562, 6.78136404695590613160172746124, 7.41445613969885487435936421338, 8.052920287226411025381601518304
