L(s) = 1 | + (−1.27 − 0.817i)3-s + (−0.415 + 0.909i)5-s + (−0.540 − 0.158i)7-s + (0.533 + 1.16i)9-s + (1.27 − 0.817i)15-s + (0.557 + 0.643i)21-s + (0.755 − 0.654i)23-s + (−0.654 − 0.755i)25-s + (0.0612 − 0.425i)27-s + (−0.118 − 0.822i)29-s + (0.368 − 0.425i)35-s + (0.797 − 1.74i)41-s + (−1.66 − 1.07i)43-s − 1.28·45-s + 1.81·47-s + ⋯ |
L(s) = 1 | + (−1.27 − 0.817i)3-s + (−0.415 + 0.909i)5-s + (−0.540 − 0.158i)7-s + (0.533 + 1.16i)9-s + (1.27 − 0.817i)15-s + (0.557 + 0.643i)21-s + (0.755 − 0.654i)23-s + (−0.654 − 0.755i)25-s + (0.0612 − 0.425i)27-s + (−0.118 − 0.822i)29-s + (0.368 − 0.425i)35-s + (0.797 − 1.74i)41-s + (−1.66 − 1.07i)43-s − 1.28·45-s + 1.81·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4605463227\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4605463227\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.415 - 0.909i)T \) |
| 23 | \( 1 + (-0.755 + 0.654i)T \) |
good | 3 | \( 1 + (1.27 + 0.817i)T + (0.415 + 0.909i)T^{2} \) |
| 7 | \( 1 + (0.540 + 0.158i)T + (0.841 + 0.540i)T^{2} \) |
| 11 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 13 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 17 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 19 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 29 | \( 1 + (0.118 + 0.822i)T + (-0.959 + 0.281i)T^{2} \) |
| 31 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 37 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 41 | \( 1 + (-0.797 + 1.74i)T + (-0.654 - 0.755i)T^{2} \) |
| 43 | \( 1 + (1.66 + 1.07i)T + (0.415 + 0.909i)T^{2} \) |
| 47 | \( 1 - 1.81T + T^{2} \) |
| 53 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 59 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 61 | \( 1 + (-0.239 + 0.153i)T + (0.415 - 0.909i)T^{2} \) |
| 67 | \( 1 + (0.708 + 0.817i)T + (-0.142 + 0.989i)T^{2} \) |
| 71 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 73 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 79 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 83 | \( 1 + (0.449 + 0.983i)T + (-0.654 + 0.755i)T^{2} \) |
| 89 | \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \) |
| 97 | \( 1 + (0.654 + 0.755i)T^{2} \) |
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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
−9.298677403112354851410884792298, −8.232457501699396429875199219855, −7.21110273430466631598583575316, −6.94616900425860531895597955196, −6.13965139524490569941579063979, −5.47319054006383049613003095165, −4.31651081571803439537547471076, −3.27023232111438808093137298952, −2.09830792730490610564465291617, −0.46179538201442682120106401467,
1.18638156543007142538581776252, 3.09386655938849796228921046260, 4.13500363950947380310600543567, 4.83672101005339715288017509109, 5.50127732279376345528478399992, 6.23099717722052933928284002506, 7.17462736308209011337170663113, 8.178519681905919614793399409675, 9.102393776595161023767640860912, 9.664818417623991604996740741473