L(s) = 1 | + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.500 + 0.866i)8-s + (1.76 − 0.642i)11-s + (−0.939 + 0.342i)16-s + (−0.173 − 0.300i)17-s + (−0.766 + 1.32i)19-s + (1.76 + 0.642i)22-s + (0.766 + 0.642i)25-s + (−0.939 − 0.342i)32-s + (0.0603 − 0.342i)34-s + (−1.43 + 0.524i)38-s + (1.17 − 0.984i)41-s + (−0.326 + 0.118i)43-s + (0.939 + 1.62i)44-s + ⋯ |
L(s) = 1 | + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.500 + 0.866i)8-s + (1.76 − 0.642i)11-s + (−0.939 + 0.342i)16-s + (−0.173 − 0.300i)17-s + (−0.766 + 1.32i)19-s + (1.76 + 0.642i)22-s + (0.766 + 0.642i)25-s + (−0.939 − 0.342i)32-s + (0.0603 − 0.342i)34-s + (−1.43 + 0.524i)38-s + (1.17 − 0.984i)41-s + (−0.326 + 0.118i)43-s + (0.939 + 1.62i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.840355629\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.840355629\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 7 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 11 | \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \) |
| 13 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 17 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 31 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \) |
| 43 | \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \) |
| 47 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 67 | \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 83 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \) |
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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
−9.159739693549131109707095787588, −8.778703544635834398219920158212, −7.85639498936489857122921425386, −7.01025043519109213283599831408, −6.28190937094689100728194455062, −5.74634909115933423064297326912, −4.59686298497787781661486798272, −3.87622903704315979964898990198, −3.11273884170246690637738621808, −1.66721350363196016762576604878,
1.24723580579280442291750738281, 2.32717730817693094640086794488, 3.38187775229637607126614067051, 4.42809431644361422060776963832, 4.73040829549630836419242269738, 6.22202334319732281204330905259, 6.48050183949201136845392146948, 7.42517021186805648323445848730, 8.805466892130110390861240376707, 9.244756366162915203221774828960