L(s) = 1 | − 2i·2-s − 2·4-s + 2i·7-s − 2·11-s + i·13-s + 4·14-s − 4·16-s − 2i·17-s + 4i·22-s − 9i·23-s + 2·26-s − 4i·28-s + 5·29-s + 2·31-s + 8i·32-s + ⋯ |
L(s) = 1 | − 1.41i·2-s − 4-s + 0.755i·7-s − 0.603·11-s + 0.277i·13-s + 1.06·14-s − 16-s − 0.485i·17-s + 0.852i·22-s − 1.87i·23-s + 0.392·26-s − 0.755i·28-s + 0.928·29-s + 0.359·31-s + 1.41i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5804994260\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5804994260\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
good | 2 | \( 1 + 2iT - 2T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 9iT - 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 11iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 15T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−8.728326883587922561144396079443, −7.59855581074883485072932842942, −6.68493767190287971990430852255, −5.86523153108150287672403174493, −4.78919398410201311247615146256, −4.23602274296385177622895630489, −2.91005205076429251752636537055, −2.62866116951003123851670006487, −1.52706109683930013424901328980, −0.17806132502989746303093302123,
1.51817062995960694710678347531, 2.96315788195802517891579086132, 3.96580596026260140632211124353, 4.97142366432850910688848350470, 5.46627244284282888526708992679, 6.41535151576267958374406220776, 6.98008745326426751706722393994, 7.72669730564748187918750243858, 8.208169284579846589864487189466, 8.926425911074091246199654963462