L(s) = 1 | + i·2-s − 4-s + (−0.707 + 2.12i)5-s + 2i·7-s − i·8-s + (−2.12 − 0.707i)10-s − 4.24·11-s + 4.82i·13-s − 2·14-s + 16-s + 1.17i·17-s + 2.24·19-s + (0.707 − 2.12i)20-s − 4.24i·22-s + i·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.316 + 0.948i)5-s + 0.755i·7-s − 0.353i·8-s + (−0.670 − 0.223i)10-s − 1.27·11-s + 1.33i·13-s − 0.534·14-s + 0.250·16-s + 0.284i·17-s + 0.514·19-s + (0.158 − 0.474i)20-s − 0.904i·22-s + 0.208i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5320442977\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5320442977\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 - 2.12i)T \) |
| 23 | \( 1 - iT \) |
good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 - 4.82iT - 13T^{2} \) |
| 17 | \( 1 - 1.17iT - 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 3.41iT - 37T^{2} \) |
| 41 | \( 1 - 1.17T + 41T^{2} \) |
| 43 | \( 1 - 1.75iT - 43T^{2} \) |
| 47 | \( 1 - 4.82iT - 47T^{2} \) |
| 53 | \( 1 + 13.4iT - 53T^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 + 3.41T + 61T^{2} \) |
| 67 | \( 1 - 0.585iT - 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 3.65iT - 73T^{2} \) |
| 79 | \( 1 + 7.65T + 79T^{2} \) |
| 83 | \( 1 - 1.41iT - 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−9.531035581489414843427164931308, −8.819297536291873432578581547930, −7.87421880268275767551754607538, −7.41693228035689110172700225392, −6.52065785395316358212728379307, −5.85341102188790878535459667511, −5.00461833112262365689830399457, −4.01762862977260399731446981115, −2.99274820591295086514203060835, −2.02645531258038361027319802593,
0.20288837447339326765178753610, 1.18374612597455623702148468815, 2.62134170514927886494258234299, 3.49731474957314305465093316794, 4.48806972396487968617649106200, 5.18024510724177163766121571465, 5.88254898471465761448007439409, 7.47896331550489180335204613797, 7.76982263456732110487800925288, 8.607184266762361451559024371305