L(s) = 1 | + (−0.951 − 0.309i)4-s + (0.309 − 0.951i)5-s + (−0.809 − 1.58i)7-s + (0.587 + 0.809i)9-s + (−0.951 + 0.309i)11-s + (0.809 + 0.587i)16-s + (−1.95 + 0.309i)17-s + (−0.309 − 0.951i)19-s + (−0.587 + 0.809i)20-s + (0.221 − 0.221i)23-s + (−0.809 − 0.587i)25-s + (0.278 + 1.76i)28-s + (−1.76 + 0.278i)35-s + (−0.309 − 0.951i)36-s + (−0.221 − 0.221i)43-s + 0.999·44-s + ⋯ |
L(s) = 1 | + (−0.951 − 0.309i)4-s + (0.309 − 0.951i)5-s + (−0.809 − 1.58i)7-s + (0.587 + 0.809i)9-s + (−0.951 + 0.309i)11-s + (0.809 + 0.587i)16-s + (−1.95 + 0.309i)17-s + (−0.309 − 0.951i)19-s + (−0.587 + 0.809i)20-s + (0.221 − 0.221i)23-s + (−0.809 − 0.587i)25-s + (0.278 + 1.76i)28-s + (−1.76 + 0.278i)35-s + (−0.309 − 0.951i)36-s + (−0.221 − 0.221i)43-s + 0.999·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5239050145\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5239050145\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.309 + 0.951i)T \) |
| 11 | \( 1 + (0.951 - 0.309i)T \) |
| 19 | \( 1 + (0.309 + 0.951i)T \) |
good | 2 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 3 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 7 | \( 1 + (0.809 + 1.58i)T + (-0.587 + 0.809i)T^{2} \) |
| 13 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 17 | \( 1 + (1.95 - 0.309i)T + (0.951 - 0.309i)T^{2} \) |
| 23 | \( 1 + (-0.221 + 0.221i)T - iT^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 41 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + (0.221 + 0.221i)T + iT^{2} \) |
| 47 | \( 1 + (-0.896 + 1.76i)T + (-0.587 - 0.809i)T^{2} \) |
| 53 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 59 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-1.26 + 0.642i)T + (0.587 - 0.809i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (0.142 + 0.896i)T + (-0.951 + 0.309i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
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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
−9.864505630919948446731596162618, −9.020122223061564793885830678345, −8.271537687780031059363656582359, −7.28803399357026223412070092257, −6.47487168452936011951079010523, −5.11129962182214186144982598232, −4.59393311720421333932629227422, −3.89595447698094866251840859137, −2.09198193242867874593286432338, −0.47678583538488655542035662149,
2.36753192514365595239337081756, 3.15983365218579098004186626335, 4.21794761450044827675811404365, 5.46591977835844356016601882175, 6.16420454607233553097080627909, 6.97751055986680092530586166985, 8.142381742061286164569171801055, 8.998240823614952722179440090084, 9.482304940563954603128383715077, 10.23941222075865570839470461978