L(s) = 1 | + 2-s + (−0.5 − 0.866i)3-s + 4-s + (0.611 + 1.05i)5-s + (−0.5 − 0.866i)6-s + (1.15 + 2.38i)7-s + 8-s + (−0.499 + 0.866i)9-s + (0.611 + 1.05i)10-s + (−0.0702 − 0.121i)11-s + (−0.5 − 0.866i)12-s + (2.39 + 2.69i)13-s + (1.15 + 2.38i)14-s + (0.611 − 1.05i)15-s + 16-s + 0.186·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.288 − 0.499i)3-s + 0.5·4-s + (0.273 + 0.473i)5-s + (−0.204 − 0.353i)6-s + (0.435 + 0.900i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.193 + 0.334i)10-s + (−0.0211 − 0.0367i)11-s + (−0.144 − 0.249i)12-s + (0.663 + 0.747i)13-s + (0.307 + 0.636i)14-s + (0.157 − 0.273i)15-s + 0.250·16-s + 0.0452·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.19045 + 0.354868i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.19045 + 0.354868i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-1.15 - 2.38i)T \) |
| 13 | \( 1 + (-2.39 - 2.69i)T \) |
good | 5 | \( 1 + (-0.611 - 1.05i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.0702 + 0.121i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 0.186T + 17T^{2} \) |
| 19 | \( 1 + (0.447 - 0.775i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 0.0364T + 23T^{2} \) |
| 29 | \( 1 + (-2.99 + 5.18i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.82 - 3.15i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 0.363T + 37T^{2} \) |
| 41 | \( 1 + (-1.70 + 2.95i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.06 + 3.58i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.358 - 0.621i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.49 + 6.05i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 6.99T + 59T^{2} \) |
| 61 | \( 1 + (0.186 - 0.323i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.42 - 4.20i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (5.31 + 9.20i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.80 - 8.32i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.94 + 5.10i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 9.14T + 83T^{2} \) |
| 89 | \( 1 + 6.35T + 89T^{2} \) |
| 97 | \( 1 + (-3.24 - 5.61i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.10002477822043827621595765576, −10.22097159585024897851973614022, −8.954378705067371157316999199510, −8.101661550328299378654755827919, −6.92749361817609466602920348474, −6.20118394175731182585878176221, −5.41583665357631953454944774672, −4.26187232002925049303997346899, −2.82370342219578301511791916773, −1.76861868489595384093624899460,
1.26343247686759808623242967764, 3.13723280502099414913129093500, 4.21559989987828443064591951247, 5.04406999972493607819709474507, 5.90061984418864172318399969616, 7.00428051324115455320337470989, 8.020042383026674494087417554417, 9.040724652881459247367499099755, 10.15490811025180800512367276225, 10.82729666410424735545549111188