L(s) = 1 | + (0.939 + 0.342i)2-s + (−0.266 − 1.50i)3-s + (0.766 + 0.642i)4-s + (0.266 − 1.50i)6-s + (0.500 + 0.866i)8-s + (−1.26 + 0.460i)9-s + (0.939 + 1.62i)11-s + (0.766 − 1.32i)12-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s − 1.34·18-s + (0.326 + 1.85i)22-s + (1.17 − 0.984i)24-s + (0.173 − 0.984i)25-s + (0.266 + 0.460i)27-s + ⋯ |
L(s) = 1 | + (0.939 + 0.342i)2-s + (−0.266 − 1.50i)3-s + (0.766 + 0.642i)4-s + (0.266 − 1.50i)6-s + (0.500 + 0.866i)8-s + (−1.26 + 0.460i)9-s + (0.939 + 1.62i)11-s + (0.766 − 1.32i)12-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s − 1.34·18-s + (0.326 + 1.85i)22-s + (1.17 − 0.984i)24-s + (0.173 − 0.984i)25-s + (0.266 + 0.460i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.176136323\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.176136323\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \) |
| 5 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 17 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 53 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 59 | \( 1 + (1.76 + 0.642i)T + (0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 67 | \( 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 73 | \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \) |
| 79 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 83 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 97 | \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \) |
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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
−8.544679475265943740565289122925, −7.73717709063548751242154013417, −7.32047573159941150570473799155, −6.49889751258171344595016022084, −6.21964829309436929228184720959, −5.11723389957821482909291379196, −4.36280300032940901537312320996, −3.29945266027797467412227052557, −2.12936087948683535619722696861, −1.50046052541945540019839182565,
1.25934674060609043023673179723, 3.01477067710761075055830759071, 3.46813126143027143253076556522, 4.18113332209920085117323736000, 5.05687833599421377032916341931, 5.67200295817595962707949553802, 6.26503201638558077053712507202, 7.30048987790797324844280462833, 8.446508463984617579685226420452, 9.272271793122195410210920798728