L(s) = 1 | + (−0.415 + 0.909i)2-s + (−0.654 − 0.755i)4-s + (−0.841 + 0.540i)5-s + (0.118 − 0.822i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)10-s + (−0.544 − 1.19i)11-s + (0.239 + 1.66i)13-s + (0.698 + 0.449i)14-s + (−0.142 + 0.989i)16-s + (−0.841 + 0.540i)18-s + (1.25 + 1.45i)19-s + (0.959 + 0.281i)20-s + 1.30·22-s + (0.959 + 0.281i)23-s + ⋯ |
L(s) = 1 | + (−0.415 + 0.909i)2-s + (−0.654 − 0.755i)4-s + (−0.841 + 0.540i)5-s + (0.118 − 0.822i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)10-s + (−0.544 − 1.19i)11-s + (0.239 + 1.66i)13-s + (0.698 + 0.449i)14-s + (−0.142 + 0.989i)16-s + (−0.841 + 0.540i)18-s + (1.25 + 1.45i)19-s + (0.959 + 0.281i)20-s + 1.30·22-s + (0.959 + 0.281i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7120138332\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7120138332\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 5 | \( 1 + (0.841 - 0.540i)T \) |
| 23 | \( 1 + (-0.959 - 0.281i)T \) |
good | 3 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 7 | \( 1 + (-0.118 + 0.822i)T + (-0.959 - 0.281i)T^{2} \) |
| 11 | \( 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2} \) |
| 13 | \( 1 + (-0.239 - 1.66i)T + (-0.959 + 0.281i)T^{2} \) |
| 17 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 19 | \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \) |
| 29 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 31 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 37 | \( 1 + (-0.239 - 0.153i)T + (0.415 + 0.909i)T^{2} \) |
| 41 | \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \) |
| 43 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 47 | \( 1 - 0.284T + T^{2} \) |
| 53 | \( 1 + (-0.239 + 1.66i)T + (-0.959 - 0.281i)T^{2} \) |
| 59 | \( 1 + (-0.0405 - 0.281i)T + (-0.959 + 0.281i)T^{2} \) |
| 61 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 67 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 71 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 73 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 79 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 83 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 89 | \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \) |
| 97 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
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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
−10.37349655324038526998652975855, −9.657516604894092516316523017043, −8.510862755548202198742885532551, −7.84641491293440071304599986675, −7.16217463158190080722792467011, −6.53793961660520211888347081860, −5.30652226508433078883291986092, −4.30930825279518474277164075997, −3.49026441129056096493313685343, −1.38382651797861231223427782509,
0.990948216681463801329753978947, 2.59656417724190333240811885509, 3.53174658956549234641272534352, 4.75073032129363861225136730827, 5.26840472583165514533302748247, 7.16372633209154939809547520320, 7.63344666752142119880235585061, 8.663204434520478371182759511677, 9.249657499825763607520443008553, 10.11634986149476230898474897638