L(s) = 1 | + 2-s + (0.707 − 0.707i)3-s + 4-s + (0.620 − 2.14i)5-s + (0.707 − 0.707i)6-s + (−0.434 + 0.434i)7-s + 8-s − 1.00i·9-s + (0.620 − 2.14i)10-s − 3.27i·11-s + (0.707 − 0.707i)12-s − 6.81·13-s + (−0.434 + 0.434i)14-s + (−1.08 − 1.95i)15-s + 16-s − 7.21i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.408 − 0.408i)3-s + 0.5·4-s + (0.277 − 0.960i)5-s + (0.288 − 0.288i)6-s + (−0.164 + 0.164i)7-s + 0.353·8-s − 0.333i·9-s + (0.196 − 0.679i)10-s − 0.988i·11-s + (0.204 − 0.204i)12-s − 1.89·13-s + (−0.116 + 0.116i)14-s + (−0.278 − 0.505i)15-s + 0.250·16-s − 1.74i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0741 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0741 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.680297141\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.680297141\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.620 + 2.14i)T \) |
| 37 | \( 1 + (3.81 + 4.73i)T \) |
good | 7 | \( 1 + (0.434 - 0.434i)T - 7iT^{2} \) |
| 11 | \( 1 + 3.27iT - 11T^{2} \) |
| 13 | \( 1 + 6.81T + 13T^{2} \) |
| 17 | \( 1 + 7.21iT - 17T^{2} \) |
| 19 | \( 1 + (-4.18 - 4.18i)T + 19iT^{2} \) |
| 23 | \( 1 - 2.26T + 23T^{2} \) |
| 29 | \( 1 + (-0.945 + 0.945i)T - 29iT^{2} \) |
| 31 | \( 1 + (-3.92 - 3.92i)T + 31iT^{2} \) |
| 41 | \( 1 - 2.88iT - 41T^{2} \) |
| 43 | \( 1 - 9.33T + 43T^{2} \) |
| 47 | \( 1 + (0.889 - 0.889i)T - 47iT^{2} \) |
| 53 | \( 1 + (-9.98 - 9.98i)T + 53iT^{2} \) |
| 59 | \( 1 + (3.81 + 3.81i)T + 59iT^{2} \) |
| 61 | \( 1 + (-2.97 - 2.97i)T + 61iT^{2} \) |
| 67 | \( 1 + (3.16 + 3.16i)T + 67iT^{2} \) |
| 71 | \( 1 - 4.70T + 71T^{2} \) |
| 73 | \( 1 + (-5.38 + 5.38i)T - 73iT^{2} \) |
| 79 | \( 1 + (10.5 + 10.5i)T + 79iT^{2} \) |
| 83 | \( 1 + (5.65 + 5.65i)T + 83iT^{2} \) |
| 89 | \( 1 + (-8.28 + 8.28i)T - 89iT^{2} \) |
| 97 | \( 1 - 1.86iT - 97T^{2} \) |
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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
−9.463848748719711193131883222667, −8.937387156840965229491614811379, −7.72927497739983616687503632553, −7.28442080587629660521364916082, −6.04374912633422888403566519921, −5.26125230679904970087867666536, −4.57556174653084563496168142556, −3.18188097472656824343343866539, −2.40661963947456546830578623569, −0.880378793874569360292808470680,
2.09984771733888380199188622231, 2.82330305486097941393020733534, 3.89478833410561013409745130619, 4.79660051299367049539391557485, 5.67189515708656527870649989697, 6.98435589355158947931970932732, 7.16012729153521711592425008226, 8.299775639144786269254641056532, 9.667279574275840866927864570707, 9.970213810575561317932498941970