L(s) = 1 | + (0.903 − 1.24i)2-s + (0.951 − 0.309i)3-s + (−0.112 − 0.346i)4-s + (0.475 − 1.46i)6-s − 1.68i·7-s + (2.39 + 0.777i)8-s + (0.809 − 0.587i)9-s + (2.40 + 1.75i)11-s + (−0.214 − 0.294i)12-s + (−0.136 − 0.188i)13-s + (−2.09 − 1.52i)14-s + (3.71 − 2.70i)16-s + (−7.09 − 2.30i)17-s − 1.53i·18-s + (−0.232 + 0.716i)19-s + ⋯ |
L(s) = 1 | + (0.639 − 0.879i)2-s + (0.549 − 0.178i)3-s + (−0.0563 − 0.173i)4-s + (0.193 − 0.597i)6-s − 0.637i·7-s + (0.845 + 0.274i)8-s + (0.269 − 0.195i)9-s + (0.726 + 0.527i)11-s + (−0.0618 − 0.0851i)12-s + (−0.0379 − 0.0522i)13-s + (−0.560 − 0.407i)14-s + (0.929 − 0.675i)16-s + (−1.72 − 0.559i)17-s − 0.362i·18-s + (−0.0534 + 0.164i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.95545 - 1.29529i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.95545 - 1.29529i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.951 + 0.309i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.903 + 1.24i)T + (-0.618 - 1.90i)T^{2} \) |
| 7 | \( 1 + 1.68iT - 7T^{2} \) |
| 11 | \( 1 + (-2.40 - 1.75i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.136 + 0.188i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (7.09 + 2.30i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (0.232 - 0.716i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.512 + 0.706i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (2.12 + 6.53i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (3.03 - 9.33i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-5.95 - 8.19i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (3.07 - 2.23i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 5.27iT - 43T^{2} \) |
| 47 | \( 1 + (8.14 - 2.64i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (5.68 - 1.84i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (3.11 - 2.26i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.55 - 2.58i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (1.70 + 0.554i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (1.35 + 4.16i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (8.84 - 12.1i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.27 + 7.01i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-4.13 - 1.34i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-9.79 - 7.11i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-9.01 + 2.92i)T + (78.4 - 57.0i)T^{2} \) |
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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
−11.37727332831111681552866744511, −10.48960343807500221330568452233, −9.529113613433136410431546259597, −8.476162295104724951206320946754, −7.38345641042057828915101773526, −6.58745364557097729013120879081, −4.75072511498907717144992375409, −4.05759778213931186484258922097, −2.88260975543983214034667639423, −1.66763248784969604333322285776,
2.02602951512193184668373718993, 3.71803949187632933915169238737, 4.69375812478399269208651168807, 5.86625765997406713287300819065, 6.61564370977710051061413579215, 7.66441865967153103297886907416, 8.759689790030162708644165582928, 9.412259204135422207444386898988, 10.74162224087844141981592486298, 11.46681168256737140125757366901