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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.049988055\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.049988055\) |
\(L(\frac{3}{2})\) |
|
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
−9.296000852843757747733622175900, −8.619498440208207467402257121171, −7.79961179975624945886623842931, −6.69676303741450502161001887161, −6.16963219705657633235645357219, −5.31259092739321913687551925083, −4.50242631012420504901707838228, −3.40630118259155737007650150794, −2.20292753354796729517817336163, −0.66800668014292607595188688813,
0.821500104886940392774969281301, 2.35999444913599819707844510303, 3.49025926622925289617435377717, 4.62885209241547409491426266975, 5.43877429435754413706936530596, 6.06018293533426589488756160905, 6.99945671631161811272717919968, 7.76814689775491676438154745457, 8.667108519719853212006780283533, 9.584772620088928877764232311187