L(s) = 1 | + (−1 − i)2-s + 2i·4-s + (2 − 2i)5-s + i·7-s + (2 − 2i)8-s − 3i·9-s − 4·10-s + (1 − i)11-s + (1 − i)14-s − 4·16-s − 2·17-s + (−3 + 3i)18-s + (2 + 2i)19-s + (4 + 4i)20-s − 2·22-s + 6i·23-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + i·4-s + (0.894 − 0.894i)5-s + 0.377i·7-s + (0.707 − 0.707i)8-s − i·9-s − 1.26·10-s + (0.301 − 0.301i)11-s + (0.267 − 0.267i)14-s − 16-s − 0.485·17-s + (−0.707 + 0.707i)18-s + (0.458 + 0.458i)19-s + (0.894 + 0.894i)20-s − 0.426·22-s + 1.25i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.701323 - 0.468609i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.701323 - 0.468609i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + 3iT^{2} \) |
| 5 | \( 1 + (-2 + 2i)T - 5iT^{2} \) |
| 11 | \( 1 + (-1 + i)T - 11iT^{2} \) |
| 13 | \( 1 + 13iT^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + (-2 - 2i)T + 19iT^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + (-7 - 7i)T + 29iT^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + (5 - 5i)T - 37iT^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 + (1 - i)T - 43iT^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + (1 - i)T - 53iT^{2} \) |
| 59 | \( 1 + (-8 + 8i)T - 59iT^{2} \) |
| 61 | \( 1 + (-6 - 6i)T + 61iT^{2} \) |
| 67 | \( 1 + (-3 - 3i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + (10 + 10i)T + 83iT^{2} \) |
| 89 | \( 1 - 14iT - 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−13.10435677069817317715597291629, −12.33374275476837685879156416076, −11.41893359581473435188152530904, −10.00937462620166698547838782373, −9.197203073722881335501337628871, −8.540009028955318294445626349058, −6.84081957360369013901511394052, −5.33221791626940470964482586284, −3.49326979522568431710341003998, −1.51802328772081144901566458296,
2.23309122526353656923885425256, 4.84926899829197202780201987783, 6.27061469703395830923553716435, 7.11194844754978421779119417990, 8.332459713466063053230675288684, 9.683924366003053898879873910454, 10.42474346384492172731049067434, 11.26811707037460048297898716732, 13.21080224703493065311584299987, 14.07709379257313501434929525229