L(s) = 1 | + (0.951 − 0.309i)9-s + (0.896 − 1.76i)13-s + (0.142 − 0.896i)17-s + (−1.11 + 1.53i)29-s + (−1.58 − 0.809i)37-s + (−0.363 − 1.11i)41-s + i·49-s + (−0.0489 − 0.309i)53-s + (0.363 − 1.11i)61-s + (0.278 − 0.142i)73-s + (0.809 − 0.587i)81-s + (1.80 + 0.587i)89-s + (1.76 − 0.278i)97-s − 0.618·101-s + (1.53 − 0.5i)109-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)9-s + (0.896 − 1.76i)13-s + (0.142 − 0.896i)17-s + (−1.11 + 1.53i)29-s + (−1.58 − 0.809i)37-s + (−0.363 − 1.11i)41-s + i·49-s + (−0.0489 − 0.309i)53-s + (0.363 − 1.11i)61-s + (0.278 − 0.142i)73-s + (0.809 − 0.587i)81-s + (1.80 + 0.587i)89-s + (1.76 − 0.278i)97-s − 0.618·101-s + (1.53 − 0.5i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.393337988\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.393337988\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.896 + 1.76i)T + (-0.587 - 0.809i)T^{2} \) |
| 17 | \( 1 + (-0.142 + 0.896i)T + (-0.951 - 0.309i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 29 | \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (1.58 + 0.809i)T + (0.587 + 0.809i)T^{2} \) |
| 41 | \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 53 | \( 1 + (0.0489 + 0.309i)T + (-0.951 + 0.309i)T^{2} \) |
| 59 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 71 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.278 + 0.142i)T + (0.587 - 0.809i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 89 | \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (-1.76 + 0.278i)T + (0.951 - 0.309i)T^{2} \) |
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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
−8.599773307452145536252978837235, −7.61356939748501633282296986458, −7.22967374809209278433798129336, −6.32234070202906041104224057211, −5.44445313270051741933646964242, −4.93413493819491911179388056416, −3.64287339607775174670315998273, −3.32809014689970200764436480556, −1.96819925723893023740312552428, −0.842515910366969655842917993558,
1.48825040359220765606321554022, 2.07116336530858488929950858830, 3.55697033631481768419599102173, 4.12418432105970475336511024371, 4.84451652299861274245209553398, 5.92281261241630302735032277258, 6.56457647911929271893794644833, 7.19510400631460177604893541842, 8.047808839199457332303873745098, 8.714026072837800928149217760289