L(s) = 1 | + 2-s − 3.24·3-s + 4-s − 1.45·5-s − 3.24·6-s + 8-s + 7.50·9-s − 1.45·10-s + 4.18·11-s − 3.24·12-s + 5.22·13-s + 4.71·15-s + 16-s + 5.04·17-s + 7.50·18-s − 4.31·19-s − 1.45·20-s + 4.18·22-s − 3.03·23-s − 3.24·24-s − 2.88·25-s + 5.22·26-s − 14.5·27-s + 2.29·29-s + 4.71·30-s + 7.50·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.87·3-s + 0.5·4-s − 0.650·5-s − 1.32·6-s + 0.353·8-s + 2.50·9-s − 0.459·10-s + 1.26·11-s − 0.935·12-s + 1.44·13-s + 1.21·15-s + 0.250·16-s + 1.22·17-s + 1.76·18-s − 0.990·19-s − 0.325·20-s + 0.891·22-s − 0.631·23-s − 0.661·24-s − 0.577·25-s + 1.02·26-s − 2.80·27-s + 0.426·29-s + 0.860·30-s + 1.34·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.682300597\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.682300597\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 + 3.24T + 3T^{2} \) |
| 5 | \( 1 + 1.45T + 5T^{2} \) |
| 11 | \( 1 - 4.18T + 11T^{2} \) |
| 13 | \( 1 - 5.22T + 13T^{2} \) |
| 17 | \( 1 - 5.04T + 17T^{2} \) |
| 19 | \( 1 + 4.31T + 19T^{2} \) |
| 23 | \( 1 + 3.03T + 23T^{2} \) |
| 29 | \( 1 - 2.29T + 29T^{2} \) |
| 31 | \( 1 - 7.50T + 31T^{2} \) |
| 37 | \( 1 - 2.48T + 37T^{2} \) |
| 43 | \( 1 - 9.36T + 43T^{2} \) |
| 47 | \( 1 + 7.57T + 47T^{2} \) |
| 53 | \( 1 - 1.22T + 53T^{2} \) |
| 59 | \( 1 + 2.55T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 4.10T + 67T^{2} \) |
| 71 | \( 1 + 8.47T + 71T^{2} \) |
| 73 | \( 1 - 4.49T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 - 2.86T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−8.190905817644526112187369085238, −7.48676414864452397182749098541, −6.49512060926102484687720305289, −6.19185927462058949067200505754, −5.68513350745435948539909432297, −4.51493842755200361730051117373, −4.20026162940813773983537957816, −3.39417425863416918767212130239, −1.61622978198534610883606024422, −0.800370533483399141464316645120,
0.800370533483399141464316645120, 1.61622978198534610883606024422, 3.39417425863416918767212130239, 4.20026162940813773983537957816, 4.51493842755200361730051117373, 5.68513350745435948539909432297, 6.19185927462058949067200505754, 6.49512060926102484687720305289, 7.48676414864452397182749098541, 8.190905817644526112187369085238