L(s) = 1 | + (−0.281 + 0.959i)2-s + (−0.540 − 0.841i)3-s + (−0.841 − 0.540i)4-s + (−0.142 + 0.989i)5-s + (0.959 − 0.281i)6-s + (0.755 − 0.345i)7-s + (0.755 − 0.654i)8-s + (−0.415 + 0.909i)9-s + (−0.909 − 0.415i)10-s + i·12-s + (0.118 + 0.822i)14-s + (0.909 − 0.415i)15-s + (0.415 + 0.909i)16-s + (−0.755 − 0.654i)18-s + (0.654 − 0.755i)20-s + (−0.698 − 0.449i)21-s + ⋯ |
L(s) = 1 | + (−0.281 + 0.959i)2-s + (−0.540 − 0.841i)3-s + (−0.841 − 0.540i)4-s + (−0.142 + 0.989i)5-s + (0.959 − 0.281i)6-s + (0.755 − 0.345i)7-s + (0.755 − 0.654i)8-s + (−0.415 + 0.909i)9-s + (−0.909 − 0.415i)10-s + i·12-s + (0.118 + 0.822i)14-s + (0.909 − 0.415i)15-s + (0.415 + 0.909i)16-s + (−0.755 − 0.654i)18-s + (0.654 − 0.755i)20-s + (−0.698 − 0.449i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7532333758\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7532333758\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.281 - 0.959i)T \) |
| 3 | \( 1 + (0.540 + 0.841i)T \) |
| 5 | \( 1 + (0.142 - 0.989i)T \) |
| 23 | \( 1 + (-0.281 + 0.959i)T \) |
good | 7 | \( 1 + (-0.755 + 0.345i)T + (0.654 - 0.755i)T^{2} \) |
| 11 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 13 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 17 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 19 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 29 | \( 1 + (-1.07 - 1.66i)T + (-0.415 + 0.909i)T^{2} \) |
| 31 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 37 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 41 | \( 1 + (-1.80 - 0.258i)T + (0.959 + 0.281i)T^{2} \) |
| 43 | \( 1 + (-1.27 - 1.10i)T + (0.142 + 0.989i)T^{2} \) |
| 47 | \( 1 + 0.284iT - T^{2} \) |
| 53 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 59 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 61 | \( 1 + (-0.817 + 0.708i)T + (0.142 - 0.989i)T^{2} \) |
| 67 | \( 1 + (0.368 - 1.25i)T + (-0.841 - 0.540i)T^{2} \) |
| 71 | \( 1 + (0.841 + 0.540i)T^{2} \) |
| 73 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 79 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 83 | \( 1 + (-0.215 - 1.49i)T + (-0.959 + 0.281i)T^{2} \) |
| 89 | \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \) |
| 97 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
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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
−9.949078831209582084534239976945, −8.777215211735996060546952757889, −7.995562293233147248801159578137, −7.39459654074257212300654976158, −6.72898936790547466825239453777, −6.09023246827539512262796779761, −5.11173683206748492004367553958, −4.24413900121558454659755122206, −2.67783099432642151200313360046, −1.19382935828985861458980881642,
0.953437054036306454062667306169, 2.37974252211667679774054860327, 3.77239047338406427032562878785, 4.46241588110557540634403244436, 5.17350092850270335205640183131, 5.94204495992401865600556186412, 7.60362716944515432020738387773, 8.334588919066382693863856152829, 9.133441087743817187706884942747, 9.570744882370265738652323758004