L(s) = 1 | + 1.78·2-s − 0.0742·3-s + 1.19·4-s − 0.132·6-s + 7-s − 1.43·8-s − 2.99·9-s + 4.34·11-s − 0.0887·12-s + 5.31·13-s + 1.78·14-s − 4.96·16-s + 2.25·17-s − 5.35·18-s + 2.09·19-s − 0.0742·21-s + 7.76·22-s − 23-s + 0.106·24-s + 9.49·26-s + 0.444·27-s + 1.19·28-s − 8.32·29-s + 3.81·31-s − 5.99·32-s − 0.322·33-s + 4.03·34-s + ⋯ |
L(s) = 1 | + 1.26·2-s − 0.0428·3-s + 0.597·4-s − 0.0541·6-s + 0.377·7-s − 0.508·8-s − 0.998·9-s + 1.30·11-s − 0.0256·12-s + 1.47·13-s + 0.477·14-s − 1.24·16-s + 0.547·17-s − 1.26·18-s + 0.480·19-s − 0.0161·21-s + 1.65·22-s − 0.208·23-s + 0.0217·24-s + 1.86·26-s + 0.0856·27-s + 0.225·28-s − 1.54·29-s + 0.684·31-s − 1.05·32-s − 0.0560·33-s + 0.692·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.846751836\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.846751836\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 - 1.78T + 2T^{2} \) |
| 3 | \( 1 + 0.0742T + 3T^{2} \) |
| 11 | \( 1 - 4.34T + 11T^{2} \) |
| 13 | \( 1 - 5.31T + 13T^{2} \) |
| 17 | \( 1 - 2.25T + 17T^{2} \) |
| 19 | \( 1 - 2.09T + 19T^{2} \) |
| 29 | \( 1 + 8.32T + 29T^{2} \) |
| 31 | \( 1 - 3.81T + 31T^{2} \) |
| 37 | \( 1 - 3.34T + 37T^{2} \) |
| 41 | \( 1 - 0.389T + 41T^{2} \) |
| 43 | \( 1 - 2.40T + 43T^{2} \) |
| 47 | \( 1 + 0.385T + 47T^{2} \) |
| 53 | \( 1 - 9.38T + 53T^{2} \) |
| 59 | \( 1 + 7.25T + 59T^{2} \) |
| 61 | \( 1 - 8.11T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 3.78T + 73T^{2} \) |
| 79 | \( 1 - 5.30T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
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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
−8.564450150384013057543449766981, −7.62883431655135095967421696216, −6.60931153628496633671293562173, −5.96470244239739818253359361316, −5.57757222362214781591777506847, −4.62749405665831549897715318354, −3.72336615249092472991978150838, −3.42884314800694963558913232865, −2.22540941458293592365041621615, −0.972131520870796013393978200253,
0.972131520870796013393978200253, 2.22540941458293592365041621615, 3.42884314800694963558913232865, 3.72336615249092472991978150838, 4.62749405665831549897715318354, 5.57757222362214781591777506847, 5.96470244239739818253359361316, 6.60931153628496633671293562173, 7.62883431655135095967421696216, 8.564450150384013057543449766981