L(s) = 1 | + (−1.73 − 0.0412i)3-s + (0.707 + 0.707i)7-s + (2.99 + 0.142i)9-s − 1.76i·11-s + (−0.719 + 0.719i)13-s + (−1.55 + 1.55i)17-s + 5.12i·19-s + (−1.19 − 1.25i)21-s + (−1.47 − 1.47i)23-s + (−5.18 − 0.371i)27-s + 7.63·29-s − 0.104·31-s + (−0.0728 + 3.05i)33-s + (−0.0126 − 0.0126i)37-s + (1.27 − 1.21i)39-s + ⋯ |
L(s) = 1 | + (−0.999 − 0.0238i)3-s + (0.267 + 0.267i)7-s + (0.998 + 0.0476i)9-s − 0.532i·11-s + (−0.199 + 0.199i)13-s + (−0.376 + 0.376i)17-s + 1.17i·19-s + (−0.260 − 0.273i)21-s + (−0.306 − 0.306i)23-s + (−0.997 − 0.0714i)27-s + 1.41·29-s − 0.0187·31-s + (−0.0126 + 0.532i)33-s + (−0.00207 − 0.00207i)37-s + (0.204 − 0.194i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.154 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.154 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9530266090\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9530266090\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.73 + 0.0412i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 11 | \( 1 + 1.76iT - 11T^{2} \) |
| 13 | \( 1 + (0.719 - 0.719i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.55 - 1.55i)T - 17iT^{2} \) |
| 19 | \( 1 - 5.12iT - 19T^{2} \) |
| 23 | \( 1 + (1.47 + 1.47i)T + 23iT^{2} \) |
| 29 | \( 1 - 7.63T + 29T^{2} \) |
| 31 | \( 1 + 0.104T + 31T^{2} \) |
| 37 | \( 1 + (0.0126 + 0.0126i)T + 37iT^{2} \) |
| 41 | \( 1 + 9.35iT - 41T^{2} \) |
| 43 | \( 1 + (2.66 - 2.66i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.98 - 3.98i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.44 - 5.44i)T + 53iT^{2} \) |
| 59 | \( 1 - 5.32T + 59T^{2} \) |
| 61 | \( 1 + 4.82T + 61T^{2} \) |
| 67 | \( 1 + (3.01 + 3.01i)T + 67iT^{2} \) |
| 71 | \( 1 + 5.20iT - 71T^{2} \) |
| 73 | \( 1 + (7.10 - 7.10i)T - 73iT^{2} \) |
| 79 | \( 1 - 15.4iT - 79T^{2} \) |
| 83 | \( 1 + (-3.76 - 3.76i)T + 83iT^{2} \) |
| 89 | \( 1 - 2.98T + 89T^{2} \) |
| 97 | \( 1 + (-6.41 - 6.41i)T + 97iT^{2} \) |
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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
−9.327110222923048373709249411196, −8.434543267920273506920917104421, −7.73614780573314888889948136774, −6.72774954188661893533714318036, −6.11449847709547917126300645306, −5.38578871612729074169261100178, −4.53044518986812103600796499236, −3.68380151104240489912074390901, −2.28489348175808938032870716686, −1.10635023188404557337195130051,
0.44642152057548915560182820338, 1.74939204113180188895958141886, 3.03218828467505522891992697833, 4.41380458642335492192972537175, 4.79065769221702856925174167986, 5.71419836876780357241165262905, 6.69987273438344052925276212541, 7.12641736434240351234149052929, 8.059297216344018937840953033911, 9.005343037981399922722235348739