L(s) = 1 | + (−2.15 − 2.15i)3-s + 2.31i·7-s + 6.31i·9-s + (−3.15 + 3.15i)11-s + (−4.31 − 4.31i)13-s + 1.31·17-s + (0.158 + 0.158i)19-s + (5 − 5i)21-s + 0.316i·23-s + (7.15 − 7.15i)27-s + (2 + 2i)29-s + 2.31·31-s + 13.6·33-s + (0.683 − 0.683i)37-s + 18.6i·39-s + ⋯ |
L(s) = 1 | + (−1.24 − 1.24i)3-s + 0.875i·7-s + 2.10i·9-s + (−0.952 + 0.952i)11-s + (−1.19 − 1.19i)13-s + 0.319·17-s + (0.0363 + 0.0363i)19-s + (1.09 − 1.09i)21-s + 0.0660i·23-s + (1.37 − 1.37i)27-s + (0.371 + 0.371i)29-s + 0.416·31-s + 2.37·33-s + (0.112 − 0.112i)37-s + 2.98i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7448149850\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7448149850\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (2.15 + 2.15i)T + 3iT^{2} \) |
| 7 | \( 1 - 2.31iT - 7T^{2} \) |
| 11 | \( 1 + (3.15 - 3.15i)T - 11iT^{2} \) |
| 13 | \( 1 + (4.31 + 4.31i)T + 13iT^{2} \) |
| 17 | \( 1 - 1.31T + 17T^{2} \) |
| 19 | \( 1 + (-0.158 - 0.158i)T + 19iT^{2} \) |
| 23 | \( 1 - 0.316iT - 23T^{2} \) |
| 29 | \( 1 + (-2 - 2i)T + 29iT^{2} \) |
| 31 | \( 1 - 2.31T + 31T^{2} \) |
| 37 | \( 1 + (-0.683 + 0.683i)T - 37iT^{2} \) |
| 41 | \( 1 + 5iT - 41T^{2} \) |
| 43 | \( 1 + (-7.63 + 7.63i)T - 43iT^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + (3.31 - 3.31i)T - 53iT^{2} \) |
| 59 | \( 1 + (-1.31 + 1.31i)T - 59iT^{2} \) |
| 61 | \( 1 + (-9.63 - 9.63i)T + 61iT^{2} \) |
| 67 | \( 1 + (9.15 + 9.15i)T + 67iT^{2} \) |
| 71 | \( 1 + 8.63iT - 71T^{2} \) |
| 73 | \( 1 + 6.68iT - 73T^{2} \) |
| 79 | \( 1 + 4.31T + 79T^{2} \) |
| 83 | \( 1 + (-7.15 - 7.15i)T + 83iT^{2} \) |
| 89 | \( 1 + 3.94iT - 89T^{2} \) |
| 97 | \( 1 + 6.63T + 97T^{2} \) |
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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
−9.286691928895444772668376072397, −8.111569611119314836643051229523, −7.48704149673283759775075196555, −6.98572872660362309359265368566, −5.76236224648033828942621041757, −5.47878770274666310305277103694, −4.66775943924797378787877386985, −2.74789839054856825815862546200, −2.04783949570786613810893349096, −0.53390163103972747278135730645,
0.73080639112088721636148502379, 2.76021430837972293409226783409, 3.98707257861592397247559777689, 4.57205598899801062307428613661, 5.33628574601262472958724163697, 6.14340612147722856785704049096, 6.99306984029585002713909596574, 7.919100225552499923055426656151, 9.049268182129367501872402773732, 9.883019862293209974193747894489