L(s) = 1 | + 1.44·2-s − 1.55·3-s + 0.0881·4-s − 3.60·5-s − 2.24·6-s − 2.76·8-s − 0.582·9-s − 5.20·10-s − 4.49·11-s − 0.137·12-s − 4.13·13-s + 5.60·15-s − 4.16·16-s − 2.02·17-s − 0.841·18-s + 8.29·19-s − 0.317·20-s − 6.49·22-s − 1.06·23-s + 4.29·24-s + 7.98·25-s − 5.97·26-s + 5.57·27-s + 6.98·29-s + 8.09·30-s − 9.48·31-s − 0.498·32-s + ⋯ |
L(s) = 1 | + 1.02·2-s − 0.897·3-s + 0.0440·4-s − 1.61·5-s − 0.917·6-s − 0.976·8-s − 0.194·9-s − 1.64·10-s − 1.35·11-s − 0.0395·12-s − 1.14·13-s + 1.44·15-s − 1.04·16-s − 0.490·17-s − 0.198·18-s + 1.90·19-s − 0.0710·20-s − 1.38·22-s − 0.221·23-s + 0.876·24-s + 1.59·25-s − 1.17·26-s + 1.07·27-s + 1.29·29-s + 1.47·30-s − 1.70·31-s − 0.0880·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4167471135\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4167471135\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 - 1.44T + 2T^{2} \) |
| 3 | \( 1 + 1.55T + 3T^{2} \) |
| 5 | \( 1 + 3.60T + 5T^{2} \) |
| 11 | \( 1 + 4.49T + 11T^{2} \) |
| 13 | \( 1 + 4.13T + 13T^{2} \) |
| 17 | \( 1 + 2.02T + 17T^{2} \) |
| 19 | \( 1 - 8.29T + 19T^{2} \) |
| 23 | \( 1 + 1.06T + 23T^{2} \) |
| 29 | \( 1 - 6.98T + 29T^{2} \) |
| 31 | \( 1 + 9.48T + 31T^{2} \) |
| 37 | \( 1 + 3.74T + 37T^{2} \) |
| 43 | \( 1 + 0.515T + 43T^{2} \) |
| 47 | \( 1 + 2.75T + 47T^{2} \) |
| 53 | \( 1 + 3.50T + 53T^{2} \) |
| 59 | \( 1 - 0.933T + 59T^{2} \) |
| 61 | \( 1 - 6.59T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 - 5.20T + 71T^{2} \) |
| 73 | \( 1 - 0.219T + 73T^{2} \) |
| 79 | \( 1 - 6.21T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 + 0.841T + 97T^{2} \) |
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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
−9.089436042997955140785357843394, −8.154061920608499167104713969903, −7.48438010794385192673065596103, −6.74468152094969572763853178761, −5.51140958432480097000922858521, −5.11794426808117334530315993013, −4.49114217773277642118662473924, −3.41623856830810583787432527366, −2.76469789548241240077393575191, −0.36101717244637400149124311412,
0.36101717244637400149124311412, 2.76469789548241240077393575191, 3.41623856830810583787432527366, 4.49114217773277642118662473924, 5.11794426808117334530315993013, 5.51140958432480097000922858521, 6.74468152094969572763853178761, 7.48438010794385192673065596103, 8.154061920608499167104713969903, 9.089436042997955140785357843394