| L(s) = 1 | − 5·5-s − 30.5·7-s − 13.5·11-s − 28.0·13-s − 55.5·17-s − 27.4·19-s − 139.·23-s + 25·25-s − 178.·29-s − 297.·31-s + 152.·35-s + 159.·37-s + 140.·41-s + 5.68·43-s − 301.·47-s + 589.·49-s + 122.·53-s + 67.6·55-s − 864.·59-s − 47.6·61-s + 140.·65-s − 402.·67-s − 927.·71-s − 1.01e3·73-s + 413.·77-s − 812.·79-s + 1.38e3·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.64·7-s − 0.371·11-s − 0.598·13-s − 0.792·17-s − 0.331·19-s − 1.26·23-s + 0.200·25-s − 1.14·29-s − 1.72·31-s + 0.737·35-s + 0.709·37-s + 0.536·41-s + 0.0201·43-s − 0.937·47-s + 1.71·49-s + 0.318·53-s + 0.165·55-s − 1.90·59-s − 0.0999·61-s + 0.267·65-s − 0.733·67-s − 1.54·71-s − 1.63·73-s + 0.611·77-s − 1.15·79-s + 1.82·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.09941543237\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09941543237\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 + 30.5T + 343T^{2} \) |
| 11 | \( 1 + 13.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 28.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 55.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 27.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 139.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 178.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 297.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 159.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 140.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 5.68T + 7.95e4T^{2} \) |
| 47 | \( 1 + 301.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 122.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 864.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 47.6T + 2.26e5T^{2} \) |
| 67 | \( 1 + 402.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 927.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 812.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.38e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.42e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 124.T + 9.12e5T^{2} \) |
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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
−8.950024611807471574386985454745, −7.74881541831856549405902253691, −7.26153086542517052672025449320, −6.31356227719760248237880911823, −5.76740493894824155946934220530, −4.55804313962423107647688786277, −3.74954987079267865320915693501, −2.94782671477517389195583016794, −1.95175958902383734863131461291, −0.13245225031005361786641185040,
0.13245225031005361786641185040, 1.95175958902383734863131461291, 2.94782671477517389195583016794, 3.74954987079267865320915693501, 4.55804313962423107647688786277, 5.76740493894824155946934220530, 6.31356227719760248237880911823, 7.26153086542517052672025449320, 7.74881541831856549405902253691, 8.950024611807471574386985454745
