L(s) = 1 | + i·3-s + (−0.587 + 0.809i)5-s + (0.309 + 0.951i)7-s + (0.5 − 0.363i)11-s + (−0.951 − 0.309i)13-s + (−0.809 − 0.587i)15-s + (0.363 + 0.5i)17-s + (0.951 − 0.309i)19-s + (−0.951 + 0.309i)21-s + (−0.5 + 1.53i)23-s + i·27-s + (−1.30 − 0.951i)29-s + (−0.951 − 1.30i)31-s + (0.363 + 0.5i)33-s + (−0.951 − 0.309i)35-s + ⋯ |
L(s) = 1 | + i·3-s + (−0.587 + 0.809i)5-s + (0.309 + 0.951i)7-s + (0.5 − 0.363i)11-s + (−0.951 − 0.309i)13-s + (−0.809 − 0.587i)15-s + (0.363 + 0.5i)17-s + (0.951 − 0.309i)19-s + (−0.951 + 0.309i)21-s + (−0.5 + 1.53i)23-s + i·27-s + (−1.30 − 0.951i)29-s + (−0.951 − 1.30i)31-s + (0.363 + 0.5i)33-s + (−0.951 − 0.309i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9573646862\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9573646862\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.309 - 0.951i)T \) |
| 41 | \( 1 + (0.951 - 0.309i)T \) |
good | 3 | \( 1 - iT - T^{2} \) |
| 5 | \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 47 | \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 - 0.618iT - T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 + iT - T^{2} \) |
| 89 | \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
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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
−10.11461546199930615980998928838, −9.567867206998146406909420976364, −8.861825623103934583773289478759, −7.64199569956829276580802875132, −7.23837266657802459782848244315, −5.73170310685823106109399453819, −5.29979501293574239769257299898, −3.92701632411256822637741222932, −3.44922150668446790820367215838, −2.12432984687920650746321530622,
0.913343631389713829830884171142, 1.98748383566477736721312006568, 3.63110088906770882665329424562, 4.55199671937395314639290914545, 5.32398952590986787214099128300, 6.94807731915831566472753269104, 7.06252539669263118385714325967, 7.951468554396047307883152262429, 8.718705390508558449767906579069, 9.733168141825598783917784484550