L(s) = 1 | − 3.91·5-s − 2.64i·7-s − 5.59·13-s + 8.66i·19-s + 6i·23-s + 10.2·25-s + 10.3i·35-s − 7.00·49-s − 5.29i·59-s + 0.543·61-s + 21.8·65-s − 15.8i·71-s − 5.29i·79-s − 18.1i·83-s + 14.8i·91-s + ⋯ |
L(s) = 1 | − 1.74·5-s − 0.999i·7-s − 1.55·13-s + 1.98i·19-s + 1.25i·23-s + 2.05·25-s + 1.74i·35-s − 49-s − 0.689i·59-s + 0.0695·61-s + 2.71·65-s − 1.88i·71-s − 0.595i·79-s − 1.99i·83-s + 1.55i·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7002808940\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7002808940\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
good | 5 | \( 1 + 3.91T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 5.59T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 8.66iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 5.29iT - 59T^{2} \) |
| 61 | \( 1 - 0.543T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 15.8iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 5.29iT - 79T^{2} \) |
| 83 | \( 1 + 18.1iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−7.962978018064297255779944482036, −7.65070926236692846085440044628, −7.26238231317085325329694793310, −6.27480271769268429070952312850, −5.13517428219227846615914050264, −4.44818156719594665180013488668, −3.71915804174553497380386739838, −3.20557478807020534905098746886, −1.70668502637046193139648059305, −0.37348490182090990425567125897,
0.57968674434294991853027345425, 2.47461056492239614811562005654, 2.88079449281226174647915717800, 4.08689250090885856365995574895, 4.74248325131866230940326949396, 5.29510652037764422657480423399, 6.61031768127903509582545475608, 7.10852549827084093440482680528, 7.80727054922813629256593821187, 8.540107585821083543573483374330