L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.125 + 1.72i)3-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (1.13 − 1.31i)6-s + 0.386·7-s + (0.707 − 0.707i)8-s + (−2.96 + 0.433i)9-s − 1.00·10-s + 1.11·11-s + (−1.72 + 0.125i)12-s + (−4.63 − 4.63i)13-s + (−0.273 − 0.273i)14-s + (1.31 + 1.13i)15-s − 1.00·16-s + (0.585 − 0.585i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.0724 + 0.997i)3-s + 0.500i·4-s + (0.316 − 0.316i)5-s + (0.462 − 0.534i)6-s + 0.146·7-s + (0.250 − 0.250i)8-s + (−0.989 + 0.144i)9-s − 0.316·10-s + 0.336·11-s + (−0.498 + 0.0362i)12-s + (−1.28 − 1.28i)13-s + (−0.0731 − 0.0731i)14-s + (0.338 + 0.292i)15-s − 0.250·16-s + (0.141 − 0.141i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.257 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7414571628\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7414571628\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.125 - 1.72i)T \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
| 37 | \( 1 + (-3.10 - 5.23i)T \) |
good | 7 | \( 1 - 0.386T + 7T^{2} \) |
| 11 | \( 1 - 1.11T + 11T^{2} \) |
| 13 | \( 1 + (4.63 + 4.63i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.585 + 0.585i)T - 17iT^{2} \) |
| 19 | \( 1 + (4.45 + 4.45i)T + 19iT^{2} \) |
| 23 | \( 1 + (-3.21 + 3.21i)T - 23iT^{2} \) |
| 29 | \( 1 + (6.53 + 6.53i)T + 29iT^{2} \) |
| 31 | \( 1 + (2.95 - 2.95i)T - 31iT^{2} \) |
| 41 | \( 1 - 3.90T + 41T^{2} \) |
| 43 | \( 1 + (-2.30 - 2.30i)T + 43iT^{2} \) |
| 47 | \( 1 + 5.16iT - 47T^{2} \) |
| 53 | \( 1 - 10.1iT - 53T^{2} \) |
| 59 | \( 1 + (-7.38 + 7.38i)T - 59iT^{2} \) |
| 61 | \( 1 + (-0.777 + 0.777i)T - 61iT^{2} \) |
| 67 | \( 1 - 2.69iT - 67T^{2} \) |
| 71 | \( 1 + 7.49iT - 71T^{2} \) |
| 73 | \( 1 + 9.26iT - 73T^{2} \) |
| 79 | \( 1 + (6.39 + 6.39i)T + 79iT^{2} \) |
| 83 | \( 1 - 0.150iT - 83T^{2} \) |
| 89 | \( 1 + (9.82 + 9.82i)T + 89iT^{2} \) |
| 97 | \( 1 + (4.44 + 4.44i)T + 97iT^{2} \) |
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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
−9.643241338225770445046325397450, −9.022232619136938183701366020495, −8.235947045169503644859369109866, −7.36176152954782042347795890091, −6.08675818011925407516272442487, −5.03767958101165898609236006996, −4.40131267945377521057353042393, −3.13204459798038238442097589269, −2.29261227673108254870466818583, −0.36735699792860090702398240412,
1.55802689811656644252222264635, 2.36083006593014786291008432807, 3.92447022690609403459115632482, 5.29883150519119138339358229920, 6.07977978143429450384953334619, 7.03122178508796902294037324672, 7.34041979690070981544992475745, 8.366538580518597757055781187455, 9.202060106858927931874036651465, 9.778845438888927440199202922216