| L(s) = 1 | − 11.4·5-s − 29.8·7-s + 66.2·11-s − 39.8·13-s − 107.·17-s − 70.3·19-s − 6.91·23-s + 5.33·25-s + 36.6·29-s − 231.·31-s + 340.·35-s − 36.8·37-s − 429.·41-s + 74.3·43-s − 52.5·47-s + 546.·49-s + 288.·53-s − 756.·55-s − 783.·59-s − 439.·61-s + 454.·65-s + 218.·67-s − 790.·71-s + 1.09e3·73-s − 1.97e3·77-s + 439.·79-s − 50.8·83-s + ⋯ |
| L(s) = 1 | − 1.02·5-s − 1.61·7-s + 1.81·11-s − 0.849·13-s − 1.53·17-s − 0.849·19-s − 0.0627·23-s + 0.0426·25-s + 0.234·29-s − 1.34·31-s + 1.64·35-s − 0.163·37-s − 1.63·41-s + 0.263·43-s − 0.163·47-s + 1.59·49-s + 0.746·53-s − 1.85·55-s − 1.72·59-s − 0.921·61-s + 0.867·65-s + 0.398·67-s − 1.32·71-s + 1.76·73-s − 2.92·77-s + 0.626·79-s − 0.0672·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4140943927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4140943927\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 11.4T + 125T^{2} \) |
| 7 | \( 1 + 29.8T + 343T^{2} \) |
| 11 | \( 1 - 66.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 39.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 70.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 6.91T + 1.21e4T^{2} \) |
| 29 | \( 1 - 36.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 231.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 36.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 429.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 74.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 52.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 288.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 783.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 439.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 218.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 790.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 439.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 50.8T + 5.71e5T^{2} \) |
| 89 | \( 1 + 719.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.19e3T + 9.12e5T^{2} \) |
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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
−9.143179248185738295427713465378, −8.269347125395430467440622198152, −7.00184899083052613635432275991, −6.81853636503424890198392911940, −5.97318971595086120274068786563, −4.53251159622340434294618952189, −3.91536138503863477930811280184, −3.19054034421037702907127306856, −1.92172927318510636008418674698, −0.29031197953700373773446657125,
0.29031197953700373773446657125, 1.92172927318510636008418674698, 3.19054034421037702907127306856, 3.91536138503863477930811280184, 4.53251159622340434294618952189, 5.97318971595086120274068786563, 6.81853636503424890198392911940, 7.00184899083052613635432275991, 8.269347125395430467440622198152, 9.143179248185738295427713465378
