L(s) = 1 | + 2-s + (0.707 + 0.707i)3-s + 4-s + (−2.22 + 0.227i)5-s + (0.707 + 0.707i)6-s + 4.05i·7-s + 8-s + 1.00i·9-s + (−2.22 + 0.227i)10-s + (1.63 − 1.63i)11-s + (0.707 + 0.707i)12-s + (3.18 + 1.68i)13-s + 4.05i·14-s + (−1.73 − 1.41i)15-s + 16-s + (0.932 + 0.932i)17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.408 + 0.408i)3-s + 0.5·4-s + (−0.994 + 0.101i)5-s + (0.288 + 0.288i)6-s + 1.53i·7-s + 0.353·8-s + 0.333i·9-s + (−0.703 + 0.0720i)10-s + (0.493 − 0.493i)11-s + (0.204 + 0.204i)12-s + (0.883 + 0.467i)13-s + 1.08i·14-s + (−0.447 − 0.364i)15-s + 0.250·16-s + (0.226 + 0.226i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.73060 + 1.08975i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.73060 + 1.08975i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.22 - 0.227i)T \) |
| 13 | \( 1 + (-3.18 - 1.68i)T \) |
good | 7 | \( 1 - 4.05iT - 7T^{2} \) |
| 11 | \( 1 + (-1.63 + 1.63i)T - 11iT^{2} \) |
| 17 | \( 1 + (-0.932 - 0.932i)T + 17iT^{2} \) |
| 19 | \( 1 + (4.66 - 4.66i)T - 19iT^{2} \) |
| 23 | \( 1 + (-3.93 + 3.93i)T - 23iT^{2} \) |
| 29 | \( 1 + 6.86iT - 29T^{2} \) |
| 31 | \( 1 + (6.67 + 6.67i)T + 31iT^{2} \) |
| 37 | \( 1 + 8.04iT - 37T^{2} \) |
| 41 | \( 1 + (-6.69 - 6.69i)T + 41iT^{2} \) |
| 43 | \( 1 + (2.58 - 2.58i)T - 43iT^{2} \) |
| 47 | \( 1 - 0.559iT - 47T^{2} \) |
| 53 | \( 1 + (6.34 + 6.34i)T + 53iT^{2} \) |
| 59 | \( 1 + (2.29 + 2.29i)T + 59iT^{2} \) |
| 61 | \( 1 - 6.48T + 61T^{2} \) |
| 67 | \( 1 - 7.13T + 67T^{2} \) |
| 71 | \( 1 + (-5.56 - 5.56i)T + 71iT^{2} \) |
| 73 | \( 1 - 7.36T + 73T^{2} \) |
| 79 | \( 1 + 1.91iT - 79T^{2} \) |
| 83 | \( 1 + 0.718iT - 83T^{2} \) |
| 89 | \( 1 + (4.02 + 4.02i)T + 89iT^{2} \) |
| 97 | \( 1 + 6.60T + 97T^{2} \) |
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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
−11.38661909916693822662630838199, −11.03391154864915614117351781890, −9.516114428068631790445758782133, −8.580824804638590656474358774741, −7.991423733316624257553985626461, −6.46643075331482265219633454549, −5.70033026805611793384239646676, −4.31293856440506382695455931706, −3.55202264316065212124950670226, −2.26931558250495156595000518431,
1.19268503725986286167511322839, 3.26806712172980784583627242423, 3.97870938715301883597993808993, 5.02278392585028426515790151770, 6.82791624727879644427735451616, 7.10781350005803909051523303898, 8.153674918799452195810440808892, 9.182678261598029920972511146379, 10.75455628144459914092524819447, 11.01362235292128966576443052617