L(s) = 1 | + i·2-s − 3.26i·3-s − 4-s + (0.904 − 2.04i)5-s + 3.26·6-s − 1.22i·7-s − i·8-s − 7.63·9-s + (2.04 + 0.904i)10-s + 4.91·11-s + 3.26i·12-s + 3.77i·13-s + 1.22·14-s + (−6.66 − 2.95i)15-s + 16-s − 6.24i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 1.88i·3-s − 0.5·4-s + (0.404 − 0.914i)5-s + 1.33·6-s − 0.462i·7-s − 0.353i·8-s − 2.54·9-s + (0.646 + 0.286i)10-s + 1.48·11-s + 0.941i·12-s + 1.04i·13-s + 0.326·14-s + (−1.72 − 0.761i)15-s + 0.250·16-s − 1.51i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.404 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.703141 - 1.07995i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.703141 - 1.07995i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (-0.904 + 2.04i)T \) |
| 43 | \( 1 + iT \) |
good | 3 | \( 1 + 3.26iT - 3T^{2} \) |
| 7 | \( 1 + 1.22iT - 7T^{2} \) |
| 11 | \( 1 - 4.91T + 11T^{2} \) |
| 13 | \( 1 - 3.77iT - 13T^{2} \) |
| 17 | \( 1 + 6.24iT - 17T^{2} \) |
| 19 | \( 1 + 5.03T + 19T^{2} \) |
| 23 | \( 1 - 4.72iT - 23T^{2} \) |
| 29 | \( 1 + 4.33T + 29T^{2} \) |
| 31 | \( 1 - 1.96T + 31T^{2} \) |
| 37 | \( 1 + 4.33iT - 37T^{2} \) |
| 41 | \( 1 - 6.72T + 41T^{2} \) |
| 47 | \( 1 - 4.48iT - 47T^{2} \) |
| 53 | \( 1 + 1.85iT - 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 - 9.35T + 61T^{2} \) |
| 67 | \( 1 + 11.0iT - 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 1.14iT - 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 - 9.21iT - 83T^{2} \) |
| 89 | \( 1 - 3.91T + 89T^{2} \) |
| 97 | \( 1 + 5.00iT - 97T^{2} \) |
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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
−11.32739442669559228408859257896, −9.351021167898266728098261454236, −8.983726587448561891767226700030, −7.88767991787231349095078375001, −7.02335485972315846010459690624, −6.46222467464096960170939850530, −5.48565450909164916200483240503, −4.10623330598031073691314450573, −2.02509391645407905580753883313, −0.861770821271889521128100405313,
2.44070232165272164359186378987, 3.60681921071243685020827900725, 4.22666467516374269657825076894, 5.60477679515118539343419242614, 6.34558267437983112540814073737, 8.374391662301794372637965852149, 9.001627036107090719812580782148, 9.964677183548046907254628365637, 10.47574808944537949851486905478, 11.07572533150720786754587493013