L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.5 − 0.866i)3-s + (−0.258 − 0.965i)5-s + (0.707 − 0.707i)6-s + (0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.499i)10-s + (−0.258 + 0.965i)11-s + (−0.500 + 0.866i)14-s + (−0.707 + 0.707i)15-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.258 − 0.965i)19-s + (0.258 − 0.965i)21-s − 22-s + (−0.866 − 0.5i)23-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.5 − 0.866i)3-s + (−0.258 − 0.965i)5-s + (0.707 − 0.707i)6-s + (0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.499i)10-s + (−0.258 + 0.965i)11-s + (−0.500 + 0.866i)14-s + (−0.707 + 0.707i)15-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.258 − 0.965i)19-s + (0.258 − 0.965i)21-s − 22-s + (−0.866 − 0.5i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.216085821\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.216085821\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.707 - 0.707i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 3 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 37 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 - iT^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 - iT^{2} \) |
| 73 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 97 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
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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
−9.831485793019243294592979057017, −8.838866405799324888291807316430, −7.972981128929056056520981223293, −7.51237800041400051968893644272, −6.58144666479003366134719317804, −5.87998137731799560674600400937, −4.95441740777550764273947237966, −4.52374474167369936385697124555, −2.40607410544921922877263731024, −1.30366030796550385958362119253,
1.57031237878080652670915735767, 2.99235024046009768568421986929, 3.76761433296570823770375489454, 4.42292790004324743908494914586, 5.57126986835363221846225213840, 6.53472137231597079122865854482, 7.71741580911029635635454859667, 8.070289125157541572281835619275, 9.825705797871159970777218029102, 10.20289736878152789957027105800