L(s) = 1 | − 2.58·2-s − 3-s + 4.68·4-s + 3.63·5-s + 2.58·6-s − 6.92·8-s + 9-s − 9.40·10-s + 1.98·11-s − 4.68·12-s + 13-s − 3.63·15-s + 8.54·16-s + 0.153·17-s − 2.58·18-s − 7.49·19-s + 17.0·20-s − 5.13·22-s − 5.05·23-s + 6.92·24-s + 8.23·25-s − 2.58·26-s − 27-s + 6.44·29-s + 9.40·30-s + 8.55·31-s − 8.22·32-s + ⋯ |
L(s) = 1 | − 1.82·2-s − 0.577·3-s + 2.34·4-s + 1.62·5-s + 1.05·6-s − 2.44·8-s + 0.333·9-s − 2.97·10-s + 0.599·11-s − 1.35·12-s + 0.277·13-s − 0.939·15-s + 2.13·16-s + 0.0372·17-s − 0.609·18-s − 1.71·19-s + 3.80·20-s − 1.09·22-s − 1.05·23-s + 1.41·24-s + 1.64·25-s − 0.506·26-s − 0.192·27-s + 1.19·29-s + 1.71·30-s + 1.53·31-s − 1.45·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8627283249\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8627283249\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 5 | \( 1 - 3.63T + 5T^{2} \) |
| 11 | \( 1 - 1.98T + 11T^{2} \) |
| 17 | \( 1 - 0.153T + 17T^{2} \) |
| 19 | \( 1 + 7.49T + 19T^{2} \) |
| 23 | \( 1 + 5.05T + 23T^{2} \) |
| 29 | \( 1 - 6.44T + 29T^{2} \) |
| 31 | \( 1 - 8.55T + 31T^{2} \) |
| 37 | \( 1 + 6.58T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 0.332T + 43T^{2} \) |
| 47 | \( 1 - 8.65T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 9.83T + 59T^{2} \) |
| 61 | \( 1 + 7.09T + 61T^{2} \) |
| 67 | \( 1 + 2.34T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 0.865T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 7.50T + 83T^{2} \) |
| 89 | \( 1 - 6.23T + 89T^{2} \) |
| 97 | \( 1 + 8.18T + 97T^{2} \) |
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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
−9.210150960804704256854882753862, −8.697795669375386454707468985617, −7.86137929433578553321656838494, −6.65591110471405760990945585959, −6.38521692471736252468770655906, −5.71853543625138165542399695614, −4.34364287997399806731515616924, −2.57896994851353340265588122459, −1.86678418213591302413303846059, −0.870570088718875782032536432864,
0.870570088718875782032536432864, 1.86678418213591302413303846059, 2.57896994851353340265588122459, 4.34364287997399806731515616924, 5.71853543625138165542399695614, 6.38521692471736252468770655906, 6.65591110471405760990945585959, 7.86137929433578553321656838494, 8.697795669375386454707468985617, 9.210150960804704256854882753862