L(s) = 1 | + (−0.707 − 0.707i)3-s + (0.707 − 0.707i)7-s + 1.73i·11-s + (1.22 − 1.22i)13-s + (1.22 + 1.22i)17-s − 1.00·21-s + (−0.707 + 0.707i)27-s + i·29-s + (1.22 − 1.22i)33-s − 1.73·39-s + (0.707 − 0.707i)47-s − 1.00i·49-s − 1.73i·51-s + (1.22 + 1.22i)77-s + 1.73·79-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)3-s + (0.707 − 0.707i)7-s + 1.73i·11-s + (1.22 − 1.22i)13-s + (1.22 + 1.22i)17-s − 1.00·21-s + (−0.707 + 0.707i)27-s + i·29-s + (1.22 − 1.22i)33-s − 1.73·39-s + (0.707 − 0.707i)47-s − 1.00i·49-s − 1.73i·51-s + (1.22 + 1.22i)77-s + 1.73·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.148255146\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.148255146\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 11 | \( 1 - 1.73iT - T^{2} \) |
| 13 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 17 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - 1.73T + T^{2} \) |
| 83 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
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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
−8.708115436226259822373677416482, −7.923536639773527445089544686823, −7.38233209318265322283135473369, −6.68858889578414075727094359579, −5.81009980586617425147828987117, −5.20703524458923090743589667548, −4.14797168103869535670558810577, −3.37433637121885887865670760529, −1.76032730771815763666870140437, −1.11173552991493214519610096061,
1.13566749562630168909727942706, 2.54069948432264765321415082376, 3.61458153298136196744153102822, 4.43465418391530016696584001667, 5.38964545808349563264763626047, 5.76583083213971685507058866256, 6.52743688631090865357999770972, 7.77579978748467592124410124559, 8.359654362961024804893279537660, 9.121943574424383410886309021695