L(s) = 1 | + (−0.309 + 0.951i)2-s + (0.743 − 0.540i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (0.284 + 0.874i)6-s + (0.809 + 0.587i)7-s + (0.809 − 0.587i)8-s + (−0.665 + 2.04i)9-s + 0.999·10-s + (3.14 − 1.05i)11-s − 0.919·12-s + (1.15 − 3.54i)13-s + (−0.809 + 0.587i)14-s + (−0.743 − 0.540i)15-s + (0.309 + 0.951i)16-s + (−0.406 − 1.25i)17-s + ⋯ |
L(s) = 1 | + (−0.218 + 0.672i)2-s + (0.429 − 0.311i)3-s + (−0.404 − 0.293i)4-s + (−0.138 − 0.425i)5-s + (0.115 + 0.356i)6-s + (0.305 + 0.222i)7-s + (0.286 − 0.207i)8-s + (−0.221 + 0.683i)9-s + 0.316·10-s + (0.948 − 0.316i)11-s − 0.265·12-s + (0.319 − 0.982i)13-s + (−0.216 + 0.157i)14-s + (−0.191 − 0.139i)15-s + (0.0772 + 0.237i)16-s + (−0.0987 − 0.303i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.59797 + 0.0432256i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.59797 + 0.0432256i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (0.309 + 0.951i)T \) |
| 7 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 + (-3.14 + 1.05i)T \) |
good | 3 | \( 1 + (-0.743 + 0.540i)T + (0.927 - 2.85i)T^{2} \) |
| 13 | \( 1 + (-1.15 + 3.54i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (0.406 + 1.25i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.19 + 1.59i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + (-2.38 - 1.73i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.38 + 7.33i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.51 - 4.73i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (2.62 - 1.90i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 6.48T + 43T^{2} \) |
| 47 | \( 1 + (-7.58 + 5.50i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-4.10 + 12.6i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-2.29 - 1.66i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.446 - 1.37i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + 6.70T + 67T^{2} \) |
| 71 | \( 1 + (-2.48 - 7.64i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.07 - 0.782i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (2.38 - 7.34i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.0881 - 0.271i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + 8.14T + 89T^{2} \) |
| 97 | \( 1 + (-4.57 + 14.0i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.12185888069909691132620954682, −9.206615708161165115163546352918, −8.401441921348913965438951244666, −7.948625309708033587344429641826, −7.00173674139315160242468936878, −5.90351827149758130577189590406, −5.12804933310276832158440244677, −4.00271241223739133456588652087, −2.59980231996983988977444118340, −1.03247529145245187578732793201,
1.32078709910533543684750428666, 2.70518947386529719657560405228, 3.86308624689997678398613999300, 4.33898626478120665299255404816, 5.98580631099300143116876636676, 6.89603905861813763870725072918, 7.916534932704430454783591303344, 8.955010713842334878081784835508, 9.355466025838422059480751530371, 10.31846096664462745822752351776