L(s) = 1 | + (0.382 + 0.923i)4-s + (1.68 + 0.902i)7-s + (−1.47 − 0.448i)13-s + (−0.707 + 0.707i)16-s + (−0.831 − 1.55i)19-s + (−0.555 + 0.831i)25-s + (−0.187 + 1.90i)28-s + (0.636 − 0.425i)31-s + (1.53 − 1.26i)37-s + (0.149 + 0.360i)43-s + (1.47 + 2.21i)49-s + (−0.151 − 1.53i)52-s − 1.96·61-s + (−0.923 − 0.382i)64-s + (0.187 + 0.0569i)67-s + ⋯ |
L(s) = 1 | + (0.382 + 0.923i)4-s + (1.68 + 0.902i)7-s + (−1.47 − 0.448i)13-s + (−0.707 + 0.707i)16-s + (−0.831 − 1.55i)19-s + (−0.555 + 0.831i)25-s + (−0.187 + 1.90i)28-s + (0.636 − 0.425i)31-s + (1.53 − 1.26i)37-s + (0.149 + 0.360i)43-s + (1.47 + 2.21i)49-s + (−0.151 − 1.53i)52-s − 1.96·61-s + (−0.923 − 0.382i)64-s + (0.187 + 0.0569i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 873 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 873 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.174601651\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.174601651\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 97 | \( 1 + (-0.555 + 0.831i)T \) |
good | 2 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 5 | \( 1 + (0.555 - 0.831i)T^{2} \) |
| 7 | \( 1 + (-1.68 - 0.902i)T + (0.555 + 0.831i)T^{2} \) |
| 11 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 13 | \( 1 + (1.47 + 0.448i)T + (0.831 + 0.555i)T^{2} \) |
| 17 | \( 1 + (0.831 + 0.555i)T^{2} \) |
| 19 | \( 1 + (0.831 + 1.55i)T + (-0.555 + 0.831i)T^{2} \) |
| 23 | \( 1 + (0.980 - 0.195i)T^{2} \) |
| 29 | \( 1 + (0.980 - 0.195i)T^{2} \) |
| 31 | \( 1 + (-0.636 + 0.425i)T + (0.382 - 0.923i)T^{2} \) |
| 37 | \( 1 + (-1.53 + 1.26i)T + (0.195 - 0.980i)T^{2} \) |
| 41 | \( 1 + (-0.195 - 0.980i)T^{2} \) |
| 43 | \( 1 + (-0.149 - 0.360i)T + (-0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 53 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 59 | \( 1 + (-0.980 - 0.195i)T^{2} \) |
| 61 | \( 1 + 1.96T + T^{2} \) |
| 67 | \( 1 + (-0.187 - 0.0569i)T + (0.831 + 0.555i)T^{2} \) |
| 71 | \( 1 + (-0.195 + 0.980i)T^{2} \) |
| 73 | \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (0.617 + 0.923i)T + (-0.382 + 0.923i)T^{2} \) |
| 83 | \( 1 + (-0.555 + 0.831i)T^{2} \) |
| 89 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
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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
−10.79113881037889470115347097677, −9.386653708552690928370765302291, −8.730408312712736030206437243384, −7.74084385723075848434965038593, −7.48463624856851664682602167948, −6.13793847489178212257811125486, −4.99366016905611748692114074335, −4.38822877483185302351965633723, −2.74930528932712349599109539927, −2.10709150590364125851547344004,
1.40097538325059712507504642898, 2.34220287757922849915631445119, 4.28859248049170599807089921917, 4.78144690721236767716156081201, 5.84653616396044699292887825970, 6.85022255975039753392156907492, 7.72502997015733101884135307442, 8.346091591860330697396174916271, 9.742061762025986235146106771191, 10.23089681101037697223022650146