| L(s) = 1 | + 4.90·2-s + 16.1·4-s − 2.10·7-s + 39.7·8-s + 40.3·11-s − 61.7·13-s − 10.3·14-s + 66.5·16-s + 99.2·17-s + 134.·19-s + 197.·22-s + 76.4·23-s − 303.·26-s − 33.8·28-s + 236.·29-s + 243.·31-s + 8.27·32-s + 487.·34-s − 57.4·37-s + 660.·38-s − 411.·41-s − 102.·43-s + 649.·44-s + 375.·46-s − 260.·47-s − 338.·49-s − 994.·52-s + ⋯ |
| L(s) = 1 | + 1.73·2-s + 2.01·4-s − 0.113·7-s + 1.75·8-s + 1.10·11-s − 1.31·13-s − 0.197·14-s + 1.03·16-s + 1.41·17-s + 1.62·19-s + 1.91·22-s + 0.693·23-s − 2.28·26-s − 0.228·28-s + 1.51·29-s + 1.41·31-s + 0.0457·32-s + 2.45·34-s − 0.255·37-s + 2.81·38-s − 1.56·41-s − 0.364·43-s + 2.22·44-s + 1.20·46-s − 0.808·47-s − 0.987·49-s − 2.65·52-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
| \(L(2)\) |
\(\approx\) |
\(6.391272454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.391272454\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 4.90T + 8T^{2} \) |
| 7 | \( 1 + 2.10T + 343T^{2} \) |
| 11 | \( 1 - 40.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 61.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 99.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 134.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 76.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 236.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 243.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 57.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 411.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 102.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 260.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 75.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 39.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 675.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 601.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 222.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 297.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 95.6T + 4.93e5T^{2} \) |
| 83 | \( 1 + 3.08T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.34e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.48e3T + 9.12e5T^{2} \) |
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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
−10.16737916373366893473148652399, −9.514226644273602357868025972577, −8.051932782925341017059814143453, −7.03497522286055854268868744583, −6.41653090233559840095575847445, −5.24030777309924559523826186649, −4.75826208979694790227748056109, −3.47621699426109596930259438237, −2.85094068169713605823198896154, −1.28324109001328204163271261247,
1.28324109001328204163271261247, 2.85094068169713605823198896154, 3.47621699426109596930259438237, 4.75826208979694790227748056109, 5.24030777309924559523826186649, 6.41653090233559840095575847445, 7.03497522286055854268868744583, 8.051932782925341017059814143453, 9.514226644273602357868025972577, 10.16737916373366893473148652399
