L(s) = 1 | + i·2-s + (−0.707 − 0.707i)3-s − 4-s + (1.04 + 1.97i)5-s + (0.707 − 0.707i)6-s + 2.54·7-s − i·8-s + 1.00i·9-s + (−1.97 + 1.04i)10-s + (−1.89 − 1.89i)11-s + (0.707 + 0.707i)12-s + (1.52 + 3.26i)13-s + 2.54i·14-s + (0.656 − 2.13i)15-s + 16-s + (−1.27 − 1.27i)17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (0.468 + 0.883i)5-s + (0.288 − 0.288i)6-s + 0.960·7-s − 0.353i·8-s + 0.333i·9-s + (−0.624 + 0.331i)10-s + (−0.571 − 0.571i)11-s + (0.204 + 0.204i)12-s + (0.424 + 0.905i)13-s + 0.679i·14-s + (0.169 − 0.551i)15-s + 0.250·16-s + (−0.310 − 0.310i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.119 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.119 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.972346 + 0.862316i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.972346 + 0.862316i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-1.04 - 1.97i)T \) |
| 13 | \( 1 + (-1.52 - 3.26i)T \) |
good | 7 | \( 1 - 2.54T + 7T^{2} \) |
| 11 | \( 1 + (1.89 + 1.89i)T + 11iT^{2} \) |
| 17 | \( 1 + (1.27 + 1.27i)T + 17iT^{2} \) |
| 19 | \( 1 + (-6.13 - 6.13i)T + 19iT^{2} \) |
| 23 | \( 1 + (0.312 - 0.312i)T - 23iT^{2} \) |
| 29 | \( 1 - 10.6iT - 29T^{2} \) |
| 31 | \( 1 + (-1 + i)T - 31iT^{2} \) |
| 37 | \( 1 + 3.18T + 37T^{2} \) |
| 41 | \( 1 + (-4.77 + 4.77i)T - 41iT^{2} \) |
| 43 | \( 1 + (-5.88 + 5.88i)T - 43iT^{2} \) |
| 47 | \( 1 + 5.18T + 47T^{2} \) |
| 53 | \( 1 + (6.13 + 6.13i)T + 53iT^{2} \) |
| 59 | \( 1 + (-7.78 + 7.78i)T - 59iT^{2} \) |
| 61 | \( 1 - 6.41T + 61T^{2} \) |
| 67 | \( 1 + 11.3iT - 67T^{2} \) |
| 71 | \( 1 + (7.30 - 7.30i)T - 71iT^{2} \) |
| 73 | \( 1 + 14.5iT - 73T^{2} \) |
| 79 | \( 1 - 3.04iT - 79T^{2} \) |
| 83 | \( 1 + 3.42T + 83T^{2} \) |
| 89 | \( 1 + (2.37 - 2.37i)T - 89iT^{2} \) |
| 97 | \( 1 - 7.40iT - 97T^{2} \) |
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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
−11.36563416213686796781647656647, −10.76618672771881267624516805420, −9.681176745410641919024261895587, −8.538972461144040660915758252323, −7.59718870615311498279888001359, −6.85092006949591642582257149123, −5.80981130212861648157112275556, −5.08314315948737832658941844200, −3.47311333546592841735033770078, −1.71082687026627518456743258674,
1.03515030369852661744986314256, 2.60718869040013214402876617418, 4.34518312368736172763694444044, 5.02252572104878990333061494960, 5.88369597337753656920215140268, 7.63933610868652656109398579912, 8.507140759490000716562662085935, 9.518529342387573238480574362355, 10.18747786950673748279042635488, 11.22244718042315188125575969751