| L(s) = 1 | − 0.946·2-s − 1.10·4-s − 5-s + 2.86·7-s + 2.93·8-s + 0.946·10-s − 0.403·11-s − 4.88·13-s − 2.71·14-s − 0.569·16-s − 6.87·17-s + 4.96·19-s + 1.10·20-s + 0.381·22-s − 4.24·23-s + 25-s + 4.62·26-s − 3.16·28-s + 7.26·29-s + 7.51·31-s − 5.33·32-s + 6.50·34-s − 2.86·35-s − 1.37·37-s − 4.70·38-s − 2.93·40-s − 8.13·41-s + ⋯ |
| L(s) = 1 | − 0.669·2-s − 0.552·4-s − 0.447·5-s + 1.08·7-s + 1.03·8-s + 0.299·10-s − 0.121·11-s − 1.35·13-s − 0.725·14-s − 0.142·16-s − 1.66·17-s + 1.13·19-s + 0.247·20-s + 0.0813·22-s − 0.885·23-s + 0.200·25-s + 0.907·26-s − 0.598·28-s + 1.34·29-s + 1.34·31-s − 0.943·32-s + 1.11·34-s − 0.484·35-s − 0.225·37-s − 0.762·38-s − 0.464·40-s − 1.27·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8561490855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8561490855\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| good | 2 | \( 1 + 0.946T + 2T^{2} \) |
| 7 | \( 1 - 2.86T + 7T^{2} \) |
| 11 | \( 1 + 0.403T + 11T^{2} \) |
| 13 | \( 1 + 4.88T + 13T^{2} \) |
| 17 | \( 1 + 6.87T + 17T^{2} \) |
| 19 | \( 1 - 4.96T + 19T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 29 | \( 1 - 7.26T + 29T^{2} \) |
| 31 | \( 1 - 7.51T + 31T^{2} \) |
| 37 | \( 1 + 1.37T + 37T^{2} \) |
| 41 | \( 1 + 8.13T + 41T^{2} \) |
| 43 | \( 1 - 3.35T + 43T^{2} \) |
| 47 | \( 1 - 4.51T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 8.09T + 59T^{2} \) |
| 61 | \( 1 + 4.94T + 61T^{2} \) |
| 67 | \( 1 + 9.11T + 67T^{2} \) |
| 71 | \( 1 - 9.34T + 71T^{2} \) |
| 73 | \( 1 + 0.0512T + 73T^{2} \) |
| 79 | \( 1 - 0.244T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−8.586518059013679111536100768613, −7.71828185417273507504462932065, −7.39697009908775342749418619673, −6.40584527870389601809969498746, −5.12977808738547120459647580201, −4.72030620119178128062741943258, −4.12627966397341695352823980804, −2.77302619131822395066670409969, −1.79632366163846854585363410023, −0.59470281854348861264286521860,
0.59470281854348861264286521860, 1.79632366163846854585363410023, 2.77302619131822395066670409969, 4.12627966397341695352823980804, 4.72030620119178128062741943258, 5.12977808738547120459647580201, 6.40584527870389601809969498746, 7.39697009908775342749418619673, 7.71828185417273507504462932065, 8.586518059013679111536100768613
