L(s) = 1 | − 3.23·3-s − 14.8·5-s + 20.4·7-s − 16.5·9-s + 30.9·11-s − 45.3·13-s + 47.9·15-s − 17·17-s − 128.·19-s − 66.1·21-s + 134.·23-s + 94.8·25-s + 140.·27-s − 214.·29-s − 82.5·31-s − 100.·33-s − 303.·35-s + 350.·37-s + 146.·39-s − 226.·41-s − 126.·43-s + 245.·45-s − 172.·47-s + 75.6·49-s + 54.9·51-s − 619.·53-s − 459.·55-s + ⋯ |
L(s) = 1 | − 0.622·3-s − 1.32·5-s + 1.10·7-s − 0.612·9-s + 0.849·11-s − 0.968·13-s + 0.825·15-s − 0.242·17-s − 1.55·19-s − 0.687·21-s + 1.21·23-s + 0.758·25-s + 1.00·27-s − 1.37·29-s − 0.477·31-s − 0.528·33-s − 1.46·35-s + 1.55·37-s + 0.602·39-s − 0.863·41-s − 0.448·43-s + 0.812·45-s − 0.534·47-s + 0.220·49-s + 0.150·51-s − 1.60·53-s − 1.12·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.7904325745\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7904325745\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + 17T \) |
good | 3 | \( 1 + 3.23T + 27T^{2} \) |
| 5 | \( 1 + 14.8T + 125T^{2} \) |
| 7 | \( 1 - 20.4T + 343T^{2} \) |
| 11 | \( 1 - 30.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 45.3T + 2.19e3T^{2} \) |
| 19 | \( 1 + 128.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 134.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 214.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 82.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 350.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 226.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 126.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 172.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 619.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 282.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 730.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 876.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 467.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 474.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.09e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 79.7T + 5.71e5T^{2} \) |
| 89 | \( 1 - 903.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 289.T + 9.12e5T^{2} \) |
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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
−9.341533825555259655513274716401, −8.536928960668532269689002407168, −7.85866268570697744596582959619, −7.05716853041611565370322867073, −6.13651556514026567951761252638, −4.93137235843341445510955142344, −4.46427088733418445898697715522, −3.35847107336300321467410162106, −1.93768558427924082297761197893, −0.47017689774618401131974828479,
0.47017689774618401131974828479, 1.93768558427924082297761197893, 3.35847107336300321467410162106, 4.46427088733418445898697715522, 4.93137235843341445510955142344, 6.13651556514026567951761252638, 7.05716853041611565370322867073, 7.85866268570697744596582959619, 8.536928960668532269689002407168, 9.341533825555259655513274716401