| L(s) = 1 | + 1.41·3-s + 2.82·5-s + 4·7-s − 0.999·9-s + 1.41·11-s − 2.82·13-s + 4.00·15-s − 4·17-s − 7.07·19-s + 5.65·21-s + 4·23-s + 3.00·25-s − 5.65·27-s − 8.48·29-s + 8·31-s + 2.00·33-s + 11.3·35-s − 2.82·37-s − 4.00·39-s + 2·41-s + 4.24·43-s − 2.82·45-s + 9·49-s − 5.65·51-s − 2.82·53-s + 4.00·55-s − 10.0·57-s + ⋯ |
| L(s) = 1 | + 0.816·3-s + 1.26·5-s + 1.51·7-s − 0.333·9-s + 0.426·11-s − 0.784·13-s + 1.03·15-s − 0.970·17-s − 1.62·19-s + 1.23·21-s + 0.834·23-s + 0.600·25-s − 1.08·27-s − 1.57·29-s + 1.43·31-s + 0.348·33-s + 1.91·35-s − 0.464·37-s − 0.640·39-s + 0.312·41-s + 0.646·43-s − 0.421·45-s + 1.28·49-s − 0.792·51-s − 0.388·53-s + 0.539·55-s − 1.32·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.355298260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.355298260\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 7.07T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 8.48T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 2.82T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4.24T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 2.82T + 53T^{2} \) |
| 59 | \( 1 - 4.24T + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 - 4.24T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−10.91209404595769029210519404696, −9.883314653812860460528287596748, −8.936781326194524102910309356880, −8.487612586958332116151505354964, −7.39445193201164554674056408186, −6.24521647187241645309584533271, −5.20880474696415961785815476276, −4.23899392513018733142803174210, −2.48954608600981803648358007327, −1.85171982909904310984154993292,
1.85171982909904310984154993292, 2.48954608600981803648358007327, 4.23899392513018733142803174210, 5.20880474696415961785815476276, 6.24521647187241645309584533271, 7.39445193201164554674056408186, 8.487612586958332116151505354964, 8.936781326194524102910309356880, 9.883314653812860460528287596748, 10.91209404595769029210519404696
