L(s) = 1 | + (0.822 − 1.80i)3-s + (0.654 − 0.755i)5-s + (−0.909 + 0.584i)7-s + (−1.91 − 2.20i)9-s + (−0.822 − 1.80i)15-s + (0.304 + 2.11i)21-s + (−0.989 + 0.142i)23-s + (−0.142 − 0.989i)25-s + (−3.64 + 1.07i)27-s + (1.25 + 0.368i)29-s + (−0.153 + 1.07i)35-s + (1.10 − 1.27i)41-s + (0.234 − 0.512i)43-s − 2.91·45-s + 1.51·47-s + ⋯ |
L(s) = 1 | + (0.822 − 1.80i)3-s + (0.654 − 0.755i)5-s + (−0.909 + 0.584i)7-s + (−1.91 − 2.20i)9-s + (−0.822 − 1.80i)15-s + (0.304 + 2.11i)21-s + (−0.989 + 0.142i)23-s + (−0.142 − 0.989i)25-s + (−3.64 + 1.07i)27-s + (1.25 + 0.368i)29-s + (−0.153 + 1.07i)35-s + (1.10 − 1.27i)41-s + (0.234 − 0.512i)43-s − 2.91·45-s + 1.51·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.386948649\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.386948649\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (0.989 - 0.142i)T \) |
good | 3 | \( 1 + (-0.822 + 1.80i)T + (-0.654 - 0.755i)T^{2} \) |
| 7 | \( 1 + (0.909 - 0.584i)T + (0.415 - 0.909i)T^{2} \) |
| 11 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 13 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 17 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 19 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 29 | \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \) |
| 31 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 37 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 41 | \( 1 + (-1.10 + 1.27i)T + (-0.142 - 0.989i)T^{2} \) |
| 43 | \( 1 + (-0.234 + 0.512i)T + (-0.654 - 0.755i)T^{2} \) |
| 47 | \( 1 - 1.51T + T^{2} \) |
| 53 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 59 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 61 | \( 1 + (-0.797 - 1.74i)T + (-0.654 + 0.755i)T^{2} \) |
| 67 | \( 1 + (0.258 + 1.80i)T + (-0.959 + 0.281i)T^{2} \) |
| 71 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 73 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 79 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 83 | \( 1 + (-1.19 - 1.37i)T + (-0.142 + 0.989i)T^{2} \) |
| 89 | \( 1 + (0.118 - 0.258i)T + (-0.654 - 0.755i)T^{2} \) |
| 97 | \( 1 + (0.142 + 0.989i)T^{2} \) |
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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
−9.029956542066544603425727025893, −8.374246594386496673155849823258, −7.61206112331055087582445679600, −6.72918560242031877264215106197, −6.09920867849493886004957408104, −5.50042607617573326626401240571, −3.88300996726463847751019915429, −2.71464688280453470870440553500, −2.13789675300489209113731301504, −0.932987039976531232467090292022,
2.38653160162398977253734846776, 3.07095966213201059815222091820, 3.86954047846114128907173441025, 4.59642968532635804992213143656, 5.69437685622620606249614888790, 6.41061327857598736938818718760, 7.54929426134093333194263359913, 8.389921170945259176834238404087, 9.273658739986589574301382216122, 9.859024516150056126164434557868