L(s) = 1 | + (−2.73 + 2.73i)5-s + 2.44i·7-s + (−1.03 + 1.03i)11-s + (3 + 3i)13-s − 3.46·17-s + (5.27 + 5.27i)19-s + 2.82i·23-s − 9.92i·25-s + (6.19 + 6.19i)29-s − 7.34·31-s + (−6.69 − 6.69i)35-s + (0.464 − 0.464i)37-s − 4.53i·41-s + (2.44 − 2.44i)43-s + 2.82·47-s + ⋯ |
L(s) = 1 | + (−1.22 + 1.22i)5-s + 0.925i·7-s + (−0.312 + 0.312i)11-s + (0.832 + 0.832i)13-s − 0.840·17-s + (1.21 + 1.21i)19-s + 0.589i·23-s − 1.98i·25-s + (1.15 + 1.15i)29-s − 1.31·31-s + (−1.13 − 1.13i)35-s + (0.0762 − 0.0762i)37-s − 0.708i·41-s + (0.373 − 0.373i)43-s + 0.412·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9265778165\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9265778165\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.73 - 2.73i)T - 5iT^{2} \) |
| 7 | \( 1 - 2.44iT - 7T^{2} \) |
| 11 | \( 1 + (1.03 - 1.03i)T - 11iT^{2} \) |
| 13 | \( 1 + (-3 - 3i)T + 13iT^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + (-5.27 - 5.27i)T + 19iT^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 + (-6.19 - 6.19i)T + 29iT^{2} \) |
| 31 | \( 1 + 7.34T + 31T^{2} \) |
| 37 | \( 1 + (-0.464 + 0.464i)T - 37iT^{2} \) |
| 41 | \( 1 + 4.53iT - 41T^{2} \) |
| 43 | \( 1 + (-2.44 + 2.44i)T - 43iT^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + (3.26 - 3.26i)T - 53iT^{2} \) |
| 59 | \( 1 + (9.79 - 9.79i)T - 59iT^{2} \) |
| 61 | \( 1 + (-0.464 - 0.464i)T + 61iT^{2} \) |
| 67 | \( 1 + (10.5 + 10.5i)T + 67iT^{2} \) |
| 71 | \( 1 + 6.41iT - 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 2.44T + 79T^{2} \) |
| 83 | \( 1 + (3.86 + 3.86i)T + 83iT^{2} \) |
| 89 | \( 1 + 4.92iT - 89T^{2} \) |
| 97 | \( 1 - 1.07T + 97T^{2} \) |
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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
−9.183307838433925396952850099453, −8.706784097428380851767244930568, −7.65004295227604530522864527720, −7.26687415714563300341461013409, −6.36056558366970873994280369803, −5.60758033552344655661830513459, −4.44927346785620056647525835312, −3.58360420534588175048098224362, −2.92540299877419486611044125122, −1.74192111962352485336687987767,
0.37768616686863925024896562651, 1.08731063889929318421155516574, 2.90189257038087699060938300773, 3.84328661198962257042717748325, 4.53437189872939734550334756276, 5.19370550391601629422596982385, 6.29101064806501705700687173913, 7.33811148429329967867489364382, 7.84670606886784775517883183672, 8.555661507232098346492347426438