| L(s) = 1 | − 2.73·2-s − 0.524·4-s − 21.1·5-s + 25.8·7-s + 23.3·8-s + 57.7·10-s − 6.96·11-s − 70.7·14-s − 59.5·16-s − 122.·17-s − 43.1·19-s + 11.0·20-s + 19.0·22-s + 75.5·23-s + 321.·25-s − 13.5·28-s + 163.·29-s − 139.·31-s − 23.6·32-s + 335.·34-s − 546.·35-s − 2.80·37-s + 117.·38-s − 492.·40-s − 300.·41-s + 363.·43-s + 3.65·44-s + ⋯ |
| L(s) = 1 | − 0.966·2-s − 0.0655·4-s − 1.88·5-s + 1.39·7-s + 1.03·8-s + 1.82·10-s − 0.191·11-s − 1.34·14-s − 0.930·16-s − 1.75·17-s − 0.520·19-s + 0.123·20-s + 0.184·22-s + 0.684·23-s + 2.56·25-s − 0.0915·28-s + 1.04·29-s − 0.807·31-s − 0.130·32-s + 1.69·34-s − 2.63·35-s − 0.0124·37-s + 0.503·38-s − 1.94·40-s − 1.14·41-s + 1.28·43-s + 0.0125·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.73T + 8T^{2} \) |
| 5 | \( 1 + 21.1T + 125T^{2} \) |
| 7 | \( 1 - 25.8T + 343T^{2} \) |
| 11 | \( 1 + 6.96T + 1.33e3T^{2} \) |
| 17 | \( 1 + 122.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 43.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 75.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 163.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 2.80T + 5.06e4T^{2} \) |
| 41 | \( 1 + 300.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 363.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 41.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 125.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 407.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 536.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 340.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 514.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 491.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 762.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 345.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 362.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 276.T + 9.12e5T^{2} \) |
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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
−8.589794221246731490584346175129, −8.083965780669618672579970652129, −7.43660443998566754119924253662, −6.74018382927478468140226924105, −4.90432994322178171151772480608, −4.58924140254017526362731536594, −3.73226333363012192449490910029, −2.20634355613184297328335247062, −0.932744334486424127369807135665, 0,
0.932744334486424127369807135665, 2.20634355613184297328335247062, 3.73226333363012192449490910029, 4.58924140254017526362731536594, 4.90432994322178171151772480608, 6.74018382927478468140226924105, 7.43660443998566754119924253662, 8.083965780669618672579970652129, 8.589794221246731490584346175129
