| L(s) = 1 | − 2i·2-s − 4·4-s + 19i·7-s + 8i·8-s + 12·11-s + 50i·13-s + 38·14-s + 16·16-s + 126i·17-s − 29·19-s − 24i·22-s + 18i·23-s + 100·26-s − 76i·28-s − 102·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 1.02i·7-s + 0.353i·8-s + 0.328·11-s + 1.06i·13-s + 0.725·14-s + 0.250·16-s + 1.79i·17-s − 0.350·19-s − 0.232i·22-s + 0.163i·23-s + 0.754·26-s − 0.512i·28-s − 0.653·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4889822334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4889822334\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 19iT - 343T^{2} \) |
| 11 | \( 1 - 12T + 1.33e3T^{2} \) |
| 13 | \( 1 - 50iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 126iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 29T + 6.85e3T^{2} \) |
| 23 | \( 1 - 18iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 102T + 2.43e4T^{2} \) |
| 31 | \( 1 + 265T + 2.97e4T^{2} \) |
| 37 | \( 1 + 65iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 240T + 6.89e4T^{2} \) |
| 43 | \( 1 + 367iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 72iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 636iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 102T + 2.05e5T^{2} \) |
| 61 | \( 1 + 103T + 2.26e5T^{2} \) |
| 67 | \( 1 - 52iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 582T + 3.57e5T^{2} \) |
| 73 | \( 1 - 65iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 173T + 4.93e5T^{2} \) |
| 83 | \( 1 - 498iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 822T + 7.04e5T^{2} \) |
| 97 | \( 1 + 821iT - 9.12e5T^{2} \) |
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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
−9.450923894528082337905424941450, −8.932819320433013685293452877503, −8.283448628614303342131233085043, −7.13813076925910733793089691943, −6.09869942739674825851724285854, −5.45048846859072180889708184532, −4.22515641553581824438072066493, −3.57831762654531185648570920684, −2.20331208157545179201644716195, −1.64004740752937379032158414104,
0.11983186337203598265607456313, 1.10393907569220871718466853577, 2.79981347286704247038586335848, 3.84114097815169696479660210201, 4.73276300285216113003933088414, 5.57771145859889737466631658844, 6.51364825569441487172515779019, 7.50698244159261628037807643379, 7.64437264043299810797003391126, 8.948228581000001923721119038019
