L(s) = 1 | − i·2-s − 4-s + (−2 + 2i)5-s + (1 + i)7-s + i·8-s + (2 + 2i)10-s + (2 + 2i)11-s + 6·13-s + (1 − i)14-s + 16-s + (−1 + 4i)17-s + 4i·19-s + (2 − 2i)20-s + (2 − 2i)22-s + (−3 − 3i)23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (−0.894 + 0.894i)5-s + (0.377 + 0.377i)7-s + 0.353i·8-s + (0.632 + 0.632i)10-s + (0.603 + 0.603i)11-s + 1.66·13-s + (0.267 − 0.267i)14-s + 0.250·16-s + (−0.242 + 0.970i)17-s + 0.917i·19-s + (0.447 − 0.447i)20-s + (0.426 − 0.426i)22-s + (−0.625 − 0.625i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08366 + 0.229803i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08366 + 0.229803i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 17 | \( 1 + (1 - 4i)T \) |
good | 5 | \( 1 + (2 - 2i)T - 5iT^{2} \) |
| 7 | \( 1 + (-1 - i)T + 7iT^{2} \) |
| 11 | \( 1 + (-2 - 2i)T + 11iT^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + (3 + 3i)T + 23iT^{2} \) |
| 29 | \( 1 + (-2 + 2i)T - 29iT^{2} \) |
| 31 | \( 1 + (3 - 3i)T - 31iT^{2} \) |
| 37 | \( 1 + (6 - 6i)T - 37iT^{2} \) |
| 41 | \( 1 + (1 + i)T + 41iT^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 - 10T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + (-2 - 2i)T + 61iT^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + (-1 + i)T - 71iT^{2} \) |
| 73 | \( 1 + (1 - i)T - 73iT^{2} \) |
| 79 | \( 1 + (5 + 5i)T + 79iT^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + (11 - 11i)T - 97iT^{2} \) |
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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
−11.82699567271427670530295196675, −10.78582451079648955639632056849, −10.34135022542770929091647528781, −8.817969526946263001489372439612, −8.205742247527223305275223905593, −6.93495799039108633180876062431, −5.84082181817088696933743350089, −4.15050813176063097862601971021, −3.49331243258603852155308560096, −1.80396497023660261662572338557,
0.919457433327447495734086435265, 3.65017507812976024460755029739, 4.50790479011145719822752276294, 5.67921479878309132841481648321, 6.85895005253238785245726957744, 7.915371461903695213301966083299, 8.656085921386729275390739070308, 9.349288390015091112167286375674, 10.99204063501587875169903683478, 11.53383237076331353896346265809