| L(s) = 1 | − 2.56·2-s + 3·3-s − 1.43·4-s − 7.68·6-s + 25.6·7-s + 24.1·8-s + 9·9-s − 11·11-s − 4.31·12-s + 8.38·13-s − 65.7·14-s − 50.4·16-s + 8.44·17-s − 23.0·18-s + 19.5·19-s + 77.0·21-s + 28.1·22-s + 189.·23-s + 72.5·24-s − 21.4·26-s + 27·27-s − 36.9·28-s + 235.·29-s + 46.4·31-s − 64.2·32-s − 33·33-s − 21.6·34-s + ⋯ |
| L(s) = 1 | − 0.905·2-s + 0.577·3-s − 0.179·4-s − 0.522·6-s + 1.38·7-s + 1.06·8-s + 0.333·9-s − 0.301·11-s − 0.103·12-s + 0.178·13-s − 1.25·14-s − 0.787·16-s + 0.120·17-s − 0.301·18-s + 0.236·19-s + 0.800·21-s + 0.273·22-s + 1.71·23-s + 0.616·24-s − 0.162·26-s + 0.192·27-s − 0.249·28-s + 1.50·29-s + 0.268·31-s − 0.354·32-s − 0.174·33-s − 0.109·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.800344061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.800344061\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 2.56T + 8T^{2} \) |
| 7 | \( 1 - 25.6T + 343T^{2} \) |
| 13 | \( 1 - 8.38T + 2.19e3T^{2} \) |
| 17 | \( 1 - 8.44T + 4.91e3T^{2} \) |
| 19 | \( 1 - 19.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 189.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 235.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 46.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 72.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 273.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 375.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 266.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 197.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 594.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 807.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 530.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 85.8T + 3.57e5T^{2} \) |
| 73 | \( 1 - 463.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.25e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 141.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 508.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.25e3T + 9.12e5T^{2} \) |
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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
−9.715332499723769574073867931986, −8.797600284541562610944091719003, −8.289180626264257829351140731491, −7.67592336250337288795171308868, −6.72700444586381726921464279766, −5.05722618480729155643028939240, −4.64057407940908256116320601349, −3.19772767377406897957777408608, −1.81238706070187559822977093913, −0.903311801815801448647873336843,
0.903311801815801448647873336843, 1.81238706070187559822977093913, 3.19772767377406897957777408608, 4.64057407940908256116320601349, 5.05722618480729155643028939240, 6.72700444586381726921464279766, 7.67592336250337288795171308868, 8.289180626264257829351140731491, 8.797600284541562610944091719003, 9.715332499723769574073867931986
