L(s) = 1 | + 1.44·3-s − 0.898·9-s + 0.550i·11-s + 7.89i·17-s + 8.34i·19-s − 5.65·27-s + 0.797i·33-s + 12.7·41-s + 10·43-s + 7·49-s + 11.4i·51-s + 12.1i·57-s + 6i·59-s − 14.3·67-s − 13.6i·73-s + ⋯ |
L(s) = 1 | + 0.836·3-s − 0.299·9-s + 0.165i·11-s + 1.91i·17-s + 1.91i·19-s − 1.08·27-s + 0.138i·33-s + 1.99·41-s + 1.52·43-s + 49-s + 1.60i·51-s + 1.60i·57-s + 0.781i·59-s − 1.75·67-s − 1.60i·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.907299712\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.907299712\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 1.44T + 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 0.550iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 7.89iT - 17T^{2} \) |
| 19 | \( 1 - 8.34iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 12.7T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 13.6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−9.394289226250301613254076990907, −8.742384915950481917328003795252, −7.977203159571139344409820308906, −7.52343773232282066242295530663, −6.07015332757248056055328264271, −5.78898511509479424494239835075, −4.22395190518720324456351522912, −3.66744709256206539299879682834, −2.54169160458573523186853681394, −1.53663954806448441323743265768,
0.66375882600806276559657655504, 2.53551104521227321984338282045, 2.85186641827609316091127985000, 4.15345079921294696344276114530, 5.05066732546330666237271840380, 5.97084569661213222854921567087, 7.15317558683897499190492050334, 7.54626438475021572319702459089, 8.676549636472985802397198225052, 9.158545144832157480576799647968