L(s) = 1 | + i·2-s + 3-s − 4-s + i·5-s + i·6-s − 3.56i·7-s − i·8-s + 9-s − 10-s + 4.12i·11-s − 12-s + 3.56·14-s + i·15-s + 16-s − 5.12·17-s + i·18-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.447i·5-s + 0.408i·6-s − 1.34i·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s + 1.24i·11-s − 0.288·12-s + 0.951·14-s + 0.258i·15-s + 0.250·16-s − 1.24·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.223611328\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.223611328\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 3.56iT - 7T^{2} \) |
| 11 | \( 1 - 4.12iT - 11T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 + 3.56iT - 19T^{2} \) |
| 23 | \( 1 - 7.68T + 23T^{2} \) |
| 29 | \( 1 - 6.56T + 29T^{2} \) |
| 31 | \( 1 - 5.68iT - 31T^{2} \) |
| 37 | \( 1 + 4.12iT - 37T^{2} \) |
| 41 | \( 1 + 4.24iT - 41T^{2} \) |
| 43 | \( 1 + 4.56T + 43T^{2} \) |
| 47 | \( 1 + 7iT - 47T^{2} \) |
| 53 | \( 1 - 4.43T + 53T^{2} \) |
| 59 | \( 1 + 10.5iT - 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 14.2iT - 67T^{2} \) |
| 71 | \( 1 + 4.87iT - 71T^{2} \) |
| 73 | \( 1 - 15.3iT - 73T^{2} \) |
| 79 | \( 1 - 7.43T + 79T^{2} \) |
| 83 | \( 1 - 1.12iT - 83T^{2} \) |
| 89 | \( 1 + 1.80iT - 89T^{2} \) |
| 97 | \( 1 - 1.12iT - 97T^{2} \) |
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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
−8.283233506015740282708905851421, −7.15126484992869238168008240004, −7.05869094008841827426696269902, −6.66576249717029833409861421799, −5.20768369016642425881875611213, −4.59034838786938362739342949573, −3.98431483478519103440254513412, −3.04394931967130554142983011255, −2.04087513681721084741147706522, −0.73970320132050704506355619621,
0.858645403548063982460919799732, 1.97571609449950513842209319654, 2.78246978519399196428664012748, 3.34833468681168197818866761913, 4.43629201704083080913939703847, 5.08236759008525371220487648706, 5.95357372469516743035332027857, 6.55205709379180738298393033216, 7.83197955891791611790452939328, 8.417564059307741006112365172192