| L(s) = 1 | + 2.79·2-s + 5.79·4-s + 5-s − 0.208·7-s + 10.5·8-s + 2.79·10-s + 0.791·11-s − 13-s − 0.582·14-s + 17.9·16-s − 17-s + 4.79·19-s + 5.79·20-s + 2.20·22-s − 7.58·23-s − 4·25-s − 2.79·26-s − 1.20·28-s − 9·29-s − 0.582·31-s + 28.9·32-s − 2.79·34-s − 0.208·35-s + 0.582·37-s + 13.3·38-s + 10.5·40-s + 9·43-s + ⋯ |
| L(s) = 1 | + 1.97·2-s + 2.89·4-s + 0.447·5-s − 0.0788·7-s + 3.74·8-s + 0.882·10-s + 0.238·11-s − 0.277·13-s − 0.155·14-s + 4.48·16-s − 0.242·17-s + 1.09·19-s + 1.29·20-s + 0.470·22-s − 1.58·23-s − 0.800·25-s − 0.547·26-s − 0.228·28-s − 1.67·29-s − 0.104·31-s + 5.11·32-s − 0.478·34-s − 0.0352·35-s + 0.0957·37-s + 2.16·38-s + 1.67·40-s + 1.37·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1989 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1989 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.927432913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.927432913\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 2.79T + 2T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 + 0.208T + 7T^{2} \) |
| 11 | \( 1 - 0.791T + 11T^{2} \) |
| 19 | \( 1 - 4.79T + 19T^{2} \) |
| 23 | \( 1 + 7.58T + 23T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + 0.582T + 31T^{2} \) |
| 37 | \( 1 - 0.582T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 9T + 43T^{2} \) |
| 47 | \( 1 + 3.58T + 47T^{2} \) |
| 53 | \( 1 - 7.79T + 53T^{2} \) |
| 59 | \( 1 + 8.58T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + 4.41T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 8.58T + 73T^{2} \) |
| 79 | \( 1 + 8.58T + 79T^{2} \) |
| 83 | \( 1 + 4.58T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−9.409238236111293147614991943784, −7.895597361737281177936433490654, −7.37077881772501712470701174459, −6.42513051436587893262599631917, −5.78028982566473091361388393580, −5.22120476527112420991759421370, −4.16815659854820814504846634635, −3.58245666235853924064529119159, −2.48250622083927589940322463339, −1.68295121708550894339422473691,
1.68295121708550894339422473691, 2.48250622083927589940322463339, 3.58245666235853924064529119159, 4.16815659854820814504846634635, 5.22120476527112420991759421370, 5.78028982566473091361388393580, 6.42513051436587893262599631917, 7.37077881772501712470701174459, 7.895597361737281177936433490654, 9.409238236111293147614991943784
