L(s) = 1 | + (−0.987 − 0.156i)2-s + (0.951 + 0.309i)4-s + (0.453 − 0.891i)5-s + (−0.891 − 0.453i)8-s + (−0.587 + 0.809i)10-s + (−0.142 − 0.896i)13-s + (0.809 + 0.587i)16-s + (0.533 − 1.04i)17-s + (0.707 − 0.707i)20-s + (−0.587 − 0.809i)25-s + 0.907i·26-s + (−0.610 + 1.87i)29-s + (−0.707 − 0.707i)32-s + (−0.690 + 0.951i)34-s + (−0.309 + 0.0489i)37-s + ⋯ |
L(s) = 1 | + (−0.987 − 0.156i)2-s + (0.951 + 0.309i)4-s + (0.453 − 0.891i)5-s + (−0.891 − 0.453i)8-s + (−0.587 + 0.809i)10-s + (−0.142 − 0.896i)13-s + (0.809 + 0.587i)16-s + (0.533 − 1.04i)17-s + (0.707 − 0.707i)20-s + (−0.587 − 0.809i)25-s + 0.907i·26-s + (−0.610 + 1.87i)29-s + (−0.707 − 0.707i)32-s + (−0.690 + 0.951i)34-s + (−0.309 + 0.0489i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.414 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.414 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6911034233\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6911034233\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.987 + 0.156i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.453 + 0.891i)T \) |
good | 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (0.142 + 0.896i)T + (-0.951 + 0.309i)T^{2} \) |
| 17 | \( 1 + (-0.533 + 1.04i)T + (-0.587 - 0.809i)T^{2} \) |
| 19 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 29 | \( 1 + (0.610 - 1.87i)T + (-0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.309 - 0.0489i)T + (0.951 - 0.309i)T^{2} \) |
| 41 | \( 1 + (-1.04 + 1.44i)T + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 53 | \( 1 + (-0.863 - 1.69i)T + (-0.587 + 0.809i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 71 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (1.76 + 0.278i)T + (0.951 + 0.309i)T^{2} \) |
| 79 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 89 | \( 1 + (0.253 - 0.183i)T + (0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.896 - 1.76i)T + (-0.587 + 0.809i)T^{2} \) |
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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
−10.09324529877556928178865240047, −9.205603916325387422983169351423, −8.735072278801126263916119766359, −7.73294976896523857412208324487, −7.04484883139849594491409430926, −5.78617789002145494035467252361, −5.10693118563936087764015871958, −3.55070126631406898522685090386, −2.35682991258214264900688641940, −0.998881365348784198240016721766,
1.72699768750245405375806757333, 2.72936958872678141768368274003, 4.03727358542339750670248167127, 5.69737542549272542945726165205, 6.29403942940161870672039323655, 7.16162774015462786521899659855, 7.921118250491204768898845507689, 8.849262584187988701252751232010, 9.849183696263354943857226894425, 10.11518563399211990573548749385