L(s) = 1 | + (0.258 − 0.965i)3-s + (−0.866 − 0.499i)9-s − 1.73i·11-s + (−0.707 + 0.707i)17-s − i·19-s + (−0.707 + 0.707i)27-s + (−1.67 − 0.448i)33-s − 1.73i·41-s + i·49-s + (0.500 + 0.866i)51-s + (−0.965 − 0.258i)57-s + (−1.22 − 1.22i)67-s + (1.22 − 1.22i)73-s + (0.500 + 0.866i)81-s + (0.707 + 0.707i)83-s + ⋯ |
L(s) = 1 | + (0.258 − 0.965i)3-s + (−0.866 − 0.499i)9-s − 1.73i·11-s + (−0.707 + 0.707i)17-s − i·19-s + (−0.707 + 0.707i)27-s + (−1.67 − 0.448i)33-s − 1.73i·41-s + i·49-s + (0.500 + 0.866i)51-s + (−0.965 − 0.258i)57-s + (−1.22 − 1.22i)67-s + (1.22 − 1.22i)73-s + (0.500 + 0.866i)81-s + (0.707 + 0.707i)83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.082257880\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.082257880\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.258 + 0.965i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + 1.73iT - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 19 | \( 1 + iT - T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + 1.73iT - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 89 | \( 1 - 1.73T + T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−8.850404177485007152454671763733, −8.135568887071482859096374610607, −7.41070661232255403704292592986, −6.46666547298134587162365203350, −6.03735049423147350261637380582, −5.08214758809088171957654050299, −3.79909814117354294672876238443, −3.00047351197616449221700554021, −2.02698487078208416048280700495, −0.68730169378393681016313771010,
1.87459035048984399193740352005, 2.81261766276377231908123865275, 3.91742498726915109474744118968, 4.60391347844255650433252267836, 5.23211089025911025170543640587, 6.29748985925294936339782605936, 7.19682827798984144199010617904, 7.950600132166587381448719577473, 8.764935228694953270829128413347, 9.612048868464700259795294949535