L(s) = 1 | + (−1 − i)2-s + (−1.96 + 1.96i)3-s + 2i·4-s + (0.254 − 2.22i)5-s + 3.93·6-s + (1.87 + 1.87i)7-s + (2 − 2i)8-s − 4.74i·9-s + (−2.47 + 1.96i)10-s + (−3.93 − 3.93i)12-s + (−4.88 + 4.88i)13-s − 3.74i·14-s + (3.87 + 4.87i)15-s − 4·16-s + (−4.74 + 4.74i)18-s + 2.41i·19-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (−1.13 + 1.13i)3-s + i·4-s + (0.113 − 0.993i)5-s + 1.60·6-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s − 1.58i·9-s + (−0.782 + 0.622i)10-s + (−1.13 − 1.13i)12-s + (−1.35 + 1.35i)13-s − 0.999i·14-s + (0.999 + 1.25i)15-s − 16-s + (−1.11 + 1.11i)18-s + 0.552i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.203948 + 0.321318i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.203948 + 0.321318i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 5 | \( 1 + (-0.254 + 2.22i)T \) |
| 7 | \( 1 + (-1.87 - 1.87i)T \) |
good | 3 | \( 1 + (1.96 - 1.96i)T - 3iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (4.88 - 4.88i)T - 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 - 2.41iT - 19T^{2} \) |
| 23 | \( 1 + (6.74 - 6.74i)T - 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 - 10.4iT - 59T^{2} \) |
| 61 | \( 1 - 6.47T + 61T^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 + 73iT^{2} \) |
| 79 | \( 1 - 8.25iT - 79T^{2} \) |
| 83 | \( 1 + (-12.2 + 12.2i)T - 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97iT^{2} \) |
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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
−11.85752877594318138977411536887, −11.41146180890839939676215005553, −10.10648886224627930277608064344, −9.573532137981231989449297832325, −8.774553588622529115870742311648, −7.56365737628655576133914653984, −5.84998599718303763183048229582, −4.82585642517749726050328896500, −4.08117541941384295050310182594, −1.87473191363303879610551512204,
0.40447727518961596242918048311, 2.19344225615377220546993600807, 4.85823112195092454801929660205, 5.89015535993968206859346902592, 6.78873115816975860768353945666, 7.48721772365423079037987633257, 8.114249422796344418749016276499, 10.00857269745167477748475604830, 10.57236959295210381426847516476, 11.35739815101660782289373823287