| L(s) = 1 | − 2.58·2-s − 1.31·4-s + 22.8·7-s + 24.0·8-s − 11.0·11-s + 11.6·13-s − 59.1·14-s − 51.7·16-s + 10.0·17-s + 117.·19-s + 28.6·22-s + 172.·23-s − 30.0·26-s − 30.1·28-s − 178.·29-s + 140.·31-s − 59.0·32-s − 26.0·34-s − 250.·37-s − 304.·38-s − 361.·41-s + 360.·43-s + 14.6·44-s − 445.·46-s − 600.·47-s + 181.·49-s − 15.3·52-s + ⋯ |
| L(s) = 1 | − 0.913·2-s − 0.164·4-s + 1.23·7-s + 1.06·8-s − 0.303·11-s + 0.248·13-s − 1.12·14-s − 0.808·16-s + 0.143·17-s + 1.42·19-s + 0.277·22-s + 1.56·23-s − 0.226·26-s − 0.203·28-s − 1.14·29-s + 0.814·31-s − 0.326·32-s − 0.131·34-s − 1.11·37-s − 1.30·38-s − 1.37·41-s + 1.27·43-s + 0.0500·44-s − 1.42·46-s − 1.86·47-s + 0.528·49-s − 0.0409·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.331390581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.331390581\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.58T + 8T^{2} \) |
| 7 | \( 1 - 22.8T + 343T^{2} \) |
| 11 | \( 1 + 11.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 11.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 10.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 117.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 172.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 178.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 140.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 250.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 361.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 360.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 600.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 201.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 415.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 54.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 531.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 933.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 560.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 810.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 538.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 686.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 714.T + 9.12e5T^{2} \) |
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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
−9.948851852859822773168337333623, −9.204094761118969320610294897859, −8.362025792834813604170241264859, −7.75219671936543122801324475633, −6.93495508565111605109026821788, −5.30444450535478341950744418190, −4.82398475795392321011905377432, −3.42429502819280665270126596257, −1.79411936803795304656447262693, −0.826904054479456536882631703960,
0.826904054479456536882631703960, 1.79411936803795304656447262693, 3.42429502819280665270126596257, 4.82398475795392321011905377432, 5.30444450535478341950744418190, 6.93495508565111605109026821788, 7.75219671936543122801324475633, 8.362025792834813604170241264859, 9.204094761118969320610294897859, 9.948851852859822773168337333623
