L(s) = 1 | + (−1.91 + 1.10i)2-s + (0.302 − 1.70i)3-s + (1.45 − 2.51i)4-s − 5-s + (1.30 + 3.60i)6-s + (2.53 − 0.756i)7-s + 2.00i·8-s + (−2.81 − 1.03i)9-s + (1.91 − 1.10i)10-s + 1.58i·11-s + (−3.85 − 3.23i)12-s + (1.44 − 0.832i)13-s + (−4.02 + 4.25i)14-s + (−0.302 + 1.70i)15-s + (0.686 + 1.18i)16-s + (−1.59 − 2.76i)17-s + ⋯ |
L(s) = 1 | + (−1.35 + 0.782i)2-s + (0.174 − 0.984i)3-s + (0.726 − 1.25i)4-s − 0.447·5-s + (0.534 + 1.47i)6-s + (0.958 − 0.286i)7-s + 0.708i·8-s + (−0.939 − 0.343i)9-s + (0.606 − 0.350i)10-s + 0.479i·11-s + (−1.11 − 0.934i)12-s + (0.400 − 0.231i)13-s + (−1.07 + 1.13i)14-s + (−0.0780 + 0.440i)15-s + (0.171 + 0.297i)16-s + (−0.386 − 0.670i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.426450 - 0.362516i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.426450 - 0.362516i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.302 + 1.70i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-2.53 + 0.756i)T \) |
good | 2 | \( 1 + (1.91 - 1.10i)T + (1 - 1.73i)T^{2} \) |
| 11 | \( 1 - 1.58iT - 11T^{2} \) |
| 13 | \( 1 + (-1.44 + 0.832i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.59 + 2.76i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.82 + 3.36i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 8.51iT - 23T^{2} \) |
| 29 | \( 1 + (-3.79 - 2.19i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.71 + 2.72i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.21 + 9.03i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.207 - 0.358i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.14 + 3.72i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.73 - 2.99i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.301 - 0.174i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.25 + 2.18i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (9.88 - 5.70i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.42 - 9.38i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8.45iT - 71T^{2} \) |
| 73 | \( 1 + (9.85 - 5.68i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.52 - 6.10i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.33 + 7.51i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.218 + 0.378i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.10 - 1.79i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.13071389604179166164780883176, −10.56631730560435456394044076095, −8.984331692825566626841052067176, −8.568029125520705884440115709266, −7.61617927593041471252422359645, −7.04912969200634531434806569070, −6.07849919003378064891896974937, −4.44294503310841507223154155516, −2.23369179515868535310328560341, −0.62380099263006709496059919332,
1.78291517481795459114218829834, 3.30185775625281871431394067385, 4.50839354482743839658485332168, 5.91320239095355864134952391447, 7.78721177587159920551139026068, 8.419261658912122656400821215485, 9.028112935918042927766539989145, 10.07043683630638723223696419011, 10.92216417280733100261113678436, 11.32567114222624521618315754266