L(s) = 1 | + (−0.826 − 0.563i)3-s + (0.988 − 0.149i)7-s + (0.365 + 0.930i)9-s + (1.03 + 1.29i)13-s + (−0.988 + 1.71i)19-s + (−0.900 − 0.433i)21-s + (−0.988 − 0.149i)25-s + (0.222 − 0.974i)27-s + (0.955 + 1.65i)31-s + (−1.40 − 0.432i)37-s + (−0.123 − 1.64i)39-s + (0.658 + 0.317i)43-s + (0.955 − 0.294i)49-s + (1.78 − 0.858i)57-s + (−0.425 − 0.131i)61-s + ⋯ |
L(s) = 1 | + (−0.826 − 0.563i)3-s + (0.988 − 0.149i)7-s + (0.365 + 0.930i)9-s + (1.03 + 1.29i)13-s + (−0.988 + 1.71i)19-s + (−0.900 − 0.433i)21-s + (−0.988 − 0.149i)25-s + (0.222 − 0.974i)27-s + (0.955 + 1.65i)31-s + (−1.40 − 0.432i)37-s + (−0.123 − 1.64i)39-s + (0.658 + 0.317i)43-s + (0.955 − 0.294i)49-s + (1.78 − 0.858i)57-s + (−0.425 − 0.131i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9902986880\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9902986880\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.826 + 0.563i)T \) |
| 7 | \( 1 + (-0.988 + 0.149i)T \) |
good | 5 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 11 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 13 | \( 1 + (-1.03 - 1.29i)T + (-0.222 + 0.974i)T^{2} \) |
| 17 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 19 | \( 1 + (0.988 - 1.71i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 29 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 31 | \( 1 + (-0.955 - 1.65i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (1.40 + 0.432i)T + (0.826 + 0.563i)T^{2} \) |
| 41 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 43 | \( 1 + (-0.658 - 0.317i)T + (0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 53 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 59 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 61 | \( 1 + (0.425 + 0.131i)T + (0.826 + 0.563i)T^{2} \) |
| 67 | \( 1 + (-0.0747 - 0.129i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (-1.44 - 0.218i)T + (0.955 + 0.294i)T^{2} \) |
| 79 | \( 1 + (-0.955 + 1.65i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 89 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 97 | \( 1 - 1.24T + T^{2} \) |
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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
−9.051710860607140867385461896413, −8.294585336620222993905653230708, −7.74786261588409208619330898066, −6.73726940108936436985948859317, −6.21452596985345852524993240923, −5.37071916659579516108962142669, −4.47504993019369767390237853387, −3.75941069371800295266570321975, −1.98359248044605318829658378785, −1.44578204741789146594300336549,
0.826980987394034488163265169230, 2.30763411432016953646803804732, 3.58929674245981148626943761216, 4.43847633907524628944300882769, 5.17319952954941135569152579424, 5.88470601268128234290161075042, 6.60602274146204089852071055597, 7.65992715617249610972379202946, 8.405346517088265625239681305409, 9.092059517077475905075472456577