L(s) = 1 | + (−0.5 + 0.866i)3-s + (−1 − 1.73i)5-s + (−0.499 − 0.866i)9-s + (2 − 3.46i)11-s + 2·13-s + 1.99·15-s + (−3 + 5.19i)17-s + (−2 − 3.46i)19-s + (0.500 − 0.866i)25-s + 0.999·27-s − 2·29-s + (1.99 + 3.46i)33-s + (−3 − 5.19i)37-s + (−1 + 1.73i)39-s − 2·41-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (−0.447 − 0.774i)5-s + (−0.166 − 0.288i)9-s + (0.603 − 1.04i)11-s + 0.554·13-s + 0.516·15-s + (−0.727 + 1.26i)17-s + (−0.458 − 0.794i)19-s + (0.100 − 0.173i)25-s + 0.192·27-s − 0.371·29-s + (0.348 + 0.603i)33-s + (−0.493 − 0.854i)37-s + (−0.160 + 0.277i)39-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6820223858\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6820223858\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (3 - 5.19i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8 + 13.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 18T + 97T^{2} \) |
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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
−8.752076864244550689075817986574, −8.256156235302716107394549011237, −7.09330532901617981437603416994, −6.16181959119288311607727680876, −5.64810322739201402753811284843, −4.43352542860517706390067571029, −4.08793253634363500271815482629, −3.03725136147849304515119055929, −1.51572481975669967895345570004, −0.24886185022786222439582601598,
1.44706056803683941569737165584, 2.50372792709442736496260014304, 3.55719680809943734518882248158, 4.44575502710521987012213463275, 5.37447014127236160357829629877, 6.49359849011394730112873966285, 6.87045450635048356605933690787, 7.56402726499231599570365145860, 8.385784379800755807502046996465, 9.300118321211605063923040790306