L(s) = 1 | + (−0.382 − 0.923i)3-s + (1.30 + 1.30i)7-s + (−0.707 + 0.707i)9-s + (0.707 − 0.292i)13-s + (−1.30 + 0.541i)19-s + (0.707 − 1.70i)21-s + (0.707 + 0.707i)25-s + (0.923 + 0.382i)27-s − 0.765·31-s + (1.70 + 0.707i)37-s + (−0.541 − 0.541i)39-s + (−0.541 + 1.30i)43-s + 2.41i·49-s + (1 + 0.999i)57-s + (−0.292 − 0.707i)61-s + ⋯ |
L(s) = 1 | + (−0.382 − 0.923i)3-s + (1.30 + 1.30i)7-s + (−0.707 + 0.707i)9-s + (0.707 − 0.292i)13-s + (−1.30 + 0.541i)19-s + (0.707 − 1.70i)21-s + (0.707 + 0.707i)25-s + (0.923 + 0.382i)27-s − 0.765·31-s + (1.70 + 0.707i)37-s + (−0.541 − 0.541i)39-s + (−0.541 + 1.30i)43-s + 2.41i·49-s + (1 + 0.999i)57-s + (−0.292 − 0.707i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.225298453\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.225298453\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.382 + 0.923i)T \) |
good | 5 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 7 | \( 1 + (-1.30 - 1.30i)T + iT^{2} \) |
| 11 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + 0.765T + T^{2} \) |
| 37 | \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 71 | \( 1 - iT^{2} \) |
| 73 | \( 1 + (-1 + i)T - iT^{2} \) |
| 79 | \( 1 + 0.765iT - T^{2} \) |
| 83 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 + 1.41T + T^{2} \) |
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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
−8.613351480286552311731129771381, −8.182645539777322994567586466522, −7.60261226699776712865457861979, −6.45452385160577013397034904537, −5.96622035073771262403099033054, −5.21422670791467994645814252381, −4.49179474851744499286368012765, −3.05974114651057671698286123068, −2.09517378355094455193726919943, −1.38130488350925638453659598462,
0.887789533032539440528755308945, 2.24303292532619186212559181329, 3.66028767285828563451165248798, 4.28124740498994560565090715496, 4.76220741261919322360261060431, 5.71121093011620643453870735485, 6.60430840550760499705570948533, 7.33132605236286392923352648152, 8.349459784305606680634166758709, 8.718797723963604676414847655392