L(s) = 1 | + 2·2-s + 4·4-s − 15.8·5-s + 8·8-s − 31.7·10-s − 57.3·11-s − 5.69·13-s + 16·16-s − 51.8·17-s − 16.2·19-s − 63.5·20-s − 114.·22-s + 213.·23-s + 127.·25-s − 11.3·26-s + 218.·29-s + 251.·31-s + 32·32-s − 103.·34-s + 386.·37-s − 32.4·38-s − 127.·40-s − 328.·41-s − 37.5·43-s − 229.·44-s + 426.·46-s − 254.·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.42·5-s + 0.353·8-s − 1.00·10-s − 1.57·11-s − 0.121·13-s + 0.250·16-s − 0.740·17-s − 0.195·19-s − 0.711·20-s − 1.11·22-s + 1.93·23-s + 1.02·25-s − 0.0859·26-s + 1.39·29-s + 1.45·31-s + 0.176·32-s − 0.523·34-s + 1.71·37-s − 0.138·38-s − 0.502·40-s − 1.25·41-s − 0.133·43-s − 0.786·44-s + 1.36·46-s − 0.791·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.919336312\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.919336312\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 15.8T + 125T^{2} \) |
| 11 | \( 1 + 57.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 5.69T + 2.19e3T^{2} \) |
| 17 | \( 1 + 51.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 16.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 213.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 218.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 251.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 386.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 328.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 37.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + 254.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 211.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 412.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 836.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 165.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 465.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 449.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 343.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.50e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 341.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 865.T + 9.12e5T^{2} \) |
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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
−9.981487478041831987089235243144, −8.573191827183235233810501577268, −8.000464215803800459443691515934, −7.18949075342023757223628090172, −6.34833181977444668783065387084, −4.90489864555221227741914496199, −4.62381056340060406599873012446, −3.29744165442864350928071707646, −2.58429822285273829867433351551, −0.66207417989178348986228028054,
0.66207417989178348986228028054, 2.58429822285273829867433351551, 3.29744165442864350928071707646, 4.62381056340060406599873012446, 4.90489864555221227741914496199, 6.34833181977444668783065387084, 7.18949075342023757223628090172, 8.000464215803800459443691515934, 8.573191827183235233810501577268, 9.981487478041831987089235243144