L(s) = 1 | + (0.309 − 0.951i)2-s + (1.30 + 0.951i)3-s + (−0.809 − 0.587i)4-s + (0.309 − 0.951i)5-s + (1.30 − 0.951i)6-s + 7-s + (−0.809 + 0.587i)8-s + (0.500 + 1.53i)9-s + (−0.809 − 0.587i)10-s + (−0.499 − 1.53i)12-s + (−0.5 − 1.53i)13-s + (0.309 − 0.951i)14-s + (1.30 − 0.951i)15-s + (0.309 + 0.951i)16-s + 1.61·18-s + (−1.61 + 1.17i)19-s + ⋯ |
L(s) = 1 | + (0.309 − 0.951i)2-s + (1.30 + 0.951i)3-s + (−0.809 − 0.587i)4-s + (0.309 − 0.951i)5-s + (1.30 − 0.951i)6-s + 7-s + (−0.809 + 0.587i)8-s + (0.500 + 1.53i)9-s + (−0.809 − 0.587i)10-s + (−0.499 − 1.53i)12-s + (−0.5 − 1.53i)13-s + (0.309 − 0.951i)14-s + (1.30 − 0.951i)15-s + (0.309 + 0.951i)16-s + 1.61·18-s + (−1.61 + 1.17i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.898922364\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.898922364\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 + (-0.309 + 0.951i)T \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 11 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
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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
−9.727326288159822723548727621095, −8.958785130999364203364742523217, −8.163669128052940320891796771701, −7.956953735065311140518057094329, −5.76596059249084394638027910492, −5.13968165364519432561176670455, −4.28570192734059590896901041333, −3.65088752799370993645319542647, −2.48721480605046224601032165137, −1.61389730590388008597355224516,
2.03560751753510727290825106473, 2.63002491771269250211801908806, 3.99961016097886550183992273094, 4.72790293626288624988884649695, 6.23826692587339069351911449873, 6.81778327730881742997899223889, 7.31323044258625766981827533383, 8.224991445658497893504273712044, 8.737817394874207415074434028150, 9.428714427671350971169107972370