L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−2 + i)5-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (0.707 − 2.12i)10-s − 0.585i·11-s + (4 − 4i)13-s + 1.00·14-s − 1.00·16-s + (0.585 − 0.585i)17-s + 2.82i·19-s + (1.00 + 2.00i)20-s + (0.414 + 0.414i)22-s + (4.82 + 4.82i)23-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.894 + 0.447i)5-s + (−0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s + (0.223 − 0.670i)10-s − 0.176i·11-s + (1.10 − 1.10i)13-s + 0.267·14-s − 0.250·16-s + (0.142 − 0.142i)17-s + 0.648i·19-s + (0.223 + 0.447i)20-s + (0.0883 + 0.0883i)22-s + (1.00 + 1.00i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.948715 + 0.193174i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.948715 + 0.193174i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 - i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 11 | \( 1 + 0.585iT - 11T^{2} \) |
| 13 | \( 1 + (-4 + 4i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.585 + 0.585i)T - 17iT^{2} \) |
| 19 | \( 1 - 2.82iT - 19T^{2} \) |
| 23 | \( 1 + (-4.82 - 4.82i)T + 23iT^{2} \) |
| 29 | \( 1 - 0.828T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 + (-6.24 - 6.24i)T + 37iT^{2} \) |
| 41 | \( 1 - 3.17iT - 41T^{2} \) |
| 43 | \( 1 + (-6.07 + 6.07i)T - 43iT^{2} \) |
| 47 | \( 1 + (-9.24 + 9.24i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.58 + 2.58i)T + 53iT^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 9.89T + 61T^{2} \) |
| 67 | \( 1 + (-8.41 - 8.41i)T + 67iT^{2} \) |
| 71 | \( 1 + 1.17iT - 71T^{2} \) |
| 73 | \( 1 + (-7.07 + 7.07i)T - 73iT^{2} \) |
| 79 | \( 1 - 5.65iT - 79T^{2} \) |
| 83 | \( 1 + (0.828 + 0.828i)T + 83iT^{2} \) |
| 89 | \( 1 - 3.17T + 89T^{2} \) |
| 97 | \( 1 + (4.58 + 4.58i)T + 97iT^{2} \) |
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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
−10.65301883132749207915944434617, −9.810806631948365861928345303126, −8.683990708971765917490205500909, −7.987927346743529696102270542461, −7.25988681917535673205037516240, −6.31032236255564282041450638292, −5.36830266801142476087593288412, −3.95854093554322050856320098167, −3.03129177528224897245582911580, −0.906476570888264820172288205241,
0.993754232869376936238642401316, 2.63786456719835176599499616725, 3.89651375802012117561995202396, 4.66932376655048733606023765884, 6.18437178795575471796420523610, 7.15779085028621566952605844419, 8.109605047898240396814859249827, 8.989207279317772114346231041723, 9.370534574105661685746341501087, 10.91702289860776923312936267692