L(s) = 1 | + 2.48i·2-s − i·3-s − 4.19·4-s + 2.48·6-s + 1.19i·7-s − 5.46i·8-s − 9-s − 1.19·11-s + 4.19i·12-s + i·13-s − 2.97·14-s + 5.21·16-s − 6.17i·17-s − 2.48i·18-s − 6.97·19-s + ⋯ |
L(s) = 1 | + 1.76i·2-s − 0.577i·3-s − 2.09·4-s + 1.01·6-s + 0.452i·7-s − 1.93i·8-s − 0.333·9-s − 0.360·11-s + 1.21i·12-s + 0.277i·13-s − 0.796·14-s + 1.30·16-s − 1.49i·17-s − 0.586i·18-s − 1.60·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.111731 - 0.0690539i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.111731 - 0.0690539i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
good | 2 | \( 1 - 2.48iT - 2T^{2} \) |
| 7 | \( 1 - 1.19iT - 7T^{2} \) |
| 11 | \( 1 + 1.19T + 11T^{2} \) |
| 17 | \( 1 + 6.17iT - 17T^{2} \) |
| 19 | \( 1 + 6.97T + 19T^{2} \) |
| 23 | \( 1 - 4.17iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 2.97T + 31T^{2} \) |
| 37 | \( 1 + 7.78iT - 37T^{2} \) |
| 41 | \( 1 + 6.17T + 41T^{2} \) |
| 43 | \( 1 + 9.95iT - 43T^{2} \) |
| 47 | \( 1 - 1.02iT - 47T^{2} \) |
| 53 | \( 1 - 10.1iT - 53T^{2} \) |
| 59 | \( 1 + 5.37T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + 9.37iT - 67T^{2} \) |
| 71 | \( 1 + 5.19T + 71T^{2} \) |
| 73 | \( 1 + 11.9iT - 73T^{2} \) |
| 79 | \( 1 - 1.78T + 79T^{2} \) |
| 83 | \( 1 + 5.37iT - 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 1.82iT - 97T^{2} \) |
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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
−9.122731980486811315230740012853, −9.028551265913521836087276876695, −7.85004914491059140278042041109, −7.31948693479767534053386853108, −6.57565638892244278597245708981, −5.67680850964086649547778111379, −5.08393976262261890803366787068, −3.88456014434641877908589326214, −2.24072169709271762784196079828, −0.05889724693817678428696016877,
1.64750551038587228950260744757, 2.74589052912346347466346002477, 3.85096843753709715796561504672, 4.33745900979908265966801562894, 5.44908867718064415994054435779, 6.65739091274957298006745735864, 8.269399661457781007960468908097, 8.616077350901613521496146708988, 9.841532883010060604669376265918, 10.23747172699111496466814559363