L(s) = 1 | + (−0.5 + 0.866i)2-s + (−1 − 1.73i)3-s + (−0.499 − 0.866i)4-s + (−1 + 1.73i)5-s + 1.99·6-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.999 − 1.73i)10-s + (−0.5 − 0.866i)11-s + (−0.999 + 1.73i)12-s − 2·13-s + 3.99·15-s + (−0.5 + 0.866i)16-s + (−0.499 − 0.866i)18-s + (−1 + 1.73i)19-s + 1.99·20-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.577 − 0.999i)3-s + (−0.249 − 0.433i)4-s + (−0.447 + 0.774i)5-s + 0.816·6-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.316 − 0.547i)10-s + (−0.150 − 0.261i)11-s + (−0.288 + 0.499i)12-s − 0.554·13-s + 1.03·15-s + (−0.125 + 0.216i)16-s + (−0.117 − 0.204i)18-s + (−0.229 + 0.397i)19-s + 0.447·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7691971396\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7691971396\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
good | 3 | \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 12T + 43T^{2} \) |
| 47 | \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5 - 8.66i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + (-6 - 10.3i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 18T + 83T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−10.15162127456614335588445302723, −8.965511703508660890420321258391, −8.105255672730875594086235302454, −7.25822316400215267067921875143, −6.86083476069200721661233291799, −6.04830428888796741109664103242, −5.15069324339980276967040440648, −3.83169165403104546495645744047, −2.47323407831660167827795338654, −0.956570724107418190049854076104,
0.55390445976021708576146310196, 2.28446684875245789166645085976, 3.65604112789896656871695376719, 4.66727994084910739298855635168, 4.92199258564704079943606270906, 6.25133201944686583939766691088, 7.53001771158119094904028620260, 8.280772633380315411645403411256, 9.235566132530795811438169605879, 9.764401971721804061150702767473