L(s) = 1 | + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.5 + 0.866i)8-s + (−1.17 + 0.984i)11-s + (0.766 − 0.642i)16-s + (−0.939 + 1.62i)17-s + (−0.173 − 0.300i)19-s + (1.17 + 0.984i)22-s + (0.173 + 0.984i)25-s + (−0.766 − 0.642i)32-s + (1.76 + 0.642i)34-s + (−0.266 + 0.223i)38-s + (−0.0603 + 0.342i)41-s + (−1.43 + 1.20i)43-s + (0.766 − 1.32i)44-s + ⋯ |
L(s) = 1 | + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.5 + 0.866i)8-s + (−1.17 + 0.984i)11-s + (0.766 − 0.642i)16-s + (−0.939 + 1.62i)17-s + (−0.173 − 0.300i)19-s + (1.17 + 0.984i)22-s + (0.173 + 0.984i)25-s + (−0.766 − 0.642i)32-s + (1.76 + 0.642i)34-s + (−0.266 + 0.223i)38-s + (−0.0603 + 0.342i)41-s + (−1.43 + 1.20i)43-s + (0.766 − 1.32i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6055467088\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6055467088\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 7 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 11 | \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \) |
| 13 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 17 | \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 29 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 31 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 67 | \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 83 | \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \) |
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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
−9.655238934477142079803068135211, −8.767151536202309104045702131199, −8.109676995604607719767968387893, −7.35099657930622985078282032823, −6.26254485999425166395760496716, −5.13619935354431997665822605743, −4.51243200769593784976302239152, −3.53934852535729187330428183508, −2.47538976066937424056853430939, −1.63677577307350172972356075262,
0.46075950541962928712663926373, 2.40381144938195629884787005015, 3.55751829014377090533045860120, 4.73019895524261238653757005801, 5.29918662267146152225296193029, 6.17376474827710253219544581218, 6.97257557795758160766166654546, 7.68695081082566784568790061898, 8.525181908552161528171554224800, 8.950317875487768726671752831427