L(s) = 1 | + i·2-s + (0.707 + 0.707i)3-s − 4-s + (−2.11 − 2.11i)5-s + (−0.707 + 0.707i)6-s + (−3.02 + 3.02i)7-s − i·8-s + 1.00i·9-s + (2.11 − 2.11i)10-s + (−0.707 + 0.707i)11-s + (−0.707 − 0.707i)12-s − 1.12·13-s + (−3.02 − 3.02i)14-s − 2.99i·15-s + 16-s + (4.12 + 0.112i)17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (−0.945 − 0.945i)5-s + (−0.288 + 0.288i)6-s + (−1.14 + 1.14i)7-s − 0.353i·8-s + 0.333i·9-s + (0.668 − 0.668i)10-s + (−0.213 + 0.213i)11-s + (−0.204 − 0.204i)12-s − 0.310·13-s + (−0.807 − 0.807i)14-s − 0.772i·15-s + 0.250·16-s + (0.999 + 0.0272i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.771 + 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7405090719\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7405090719\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.707 - 0.707i)T \) |
| 11 | \( 1 + (0.707 - 0.707i)T \) |
| 17 | \( 1 + (-4.12 - 0.112i)T \) |
good | 5 | \( 1 + (2.11 + 2.11i)T + 5iT^{2} \) |
| 7 | \( 1 + (3.02 - 3.02i)T - 7iT^{2} \) |
| 13 | \( 1 + 1.12T + 13T^{2} \) |
| 19 | \( 1 + 5.98iT - 19T^{2} \) |
| 23 | \( 1 + (-5.70 + 5.70i)T - 23iT^{2} \) |
| 29 | \( 1 + (0.420 + 0.420i)T + 29iT^{2} \) |
| 31 | \( 1 + (2.09 + 2.09i)T + 31iT^{2} \) |
| 37 | \( 1 + (-1.97 - 1.97i)T + 37iT^{2} \) |
| 41 | \( 1 + (-4.14 + 4.14i)T - 41iT^{2} \) |
| 43 | \( 1 + 5.97iT - 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 + 13.7iT - 53T^{2} \) |
| 59 | \( 1 - 10.4iT - 59T^{2} \) |
| 61 | \( 1 + (-7.71 + 7.71i)T - 61iT^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + (-0.330 - 0.330i)T + 71iT^{2} \) |
| 73 | \( 1 + (9.32 + 9.32i)T + 73iT^{2} \) |
| 79 | \( 1 + (-3.01 + 3.01i)T - 79iT^{2} \) |
| 83 | \( 1 - 7.58iT - 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 + (1.80 + 1.80i)T + 97iT^{2} \) |
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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
−9.364674552689916543537300985210, −8.898330075263177495870696311726, −8.251996740863773578951810375155, −7.33300278315533363623924641804, −6.42985467108940329600643298870, −5.27587905494011587081083140265, −4.74974451241741049681428424831, −3.59717725373618442185484027512, −2.64461868458477656056804806847, −0.34594995751602097546743599947,
1.23439461368622535017571084682, 3.16324065468764067325612967876, 3.24301395216225395938023627942, 4.23317086435097733390397760210, 5.76534735568059657050150884859, 6.86029206332497097120807650372, 7.51304061793172368375441101422, 8.041227395017764275731342787232, 9.402659617487893904810490068271, 9.974261459077462395529324176517