L(s) = 1 | + i·3-s + 0.746·7-s − 9-s + 5.36i·11-s + 2.92i·13-s − 2.13·17-s + 1.73i·19-s + 0.746i·21-s + 7.49·23-s − i·27-s − 6.74i·29-s − 2.64·31-s − 5.36·33-s − 1.07i·37-s − 2.92·39-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.282·7-s − 0.333·9-s + 1.61i·11-s + 0.811i·13-s − 0.517·17-s + 0.397i·19-s + 0.162i·21-s + 1.56·23-s − 0.192i·27-s − 1.25i·29-s − 0.475·31-s − 0.933·33-s − 0.176i·37-s − 0.468·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.277245739\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.277245739\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 0.746T + 7T^{2} \) |
| 11 | \( 1 - 5.36iT - 11T^{2} \) |
| 13 | \( 1 - 2.92iT - 13T^{2} \) |
| 17 | \( 1 + 2.13T + 17T^{2} \) |
| 19 | \( 1 - 1.73iT - 19T^{2} \) |
| 23 | \( 1 - 7.49T + 23T^{2} \) |
| 29 | \( 1 + 6.74iT - 29T^{2} \) |
| 31 | \( 1 + 2.64T + 31T^{2} \) |
| 37 | \( 1 + 1.07iT - 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 7.44iT - 43T^{2} \) |
| 47 | \( 1 - 1.73T + 47T^{2} \) |
| 53 | \( 1 + 7.72iT - 53T^{2} \) |
| 59 | \( 1 - 6.85iT - 59T^{2} \) |
| 61 | \( 1 - 6.45iT - 61T^{2} \) |
| 67 | \( 1 - 7.44iT - 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 0.690T + 73T^{2} \) |
| 79 | \( 1 + 2.64T + 79T^{2} \) |
| 83 | \( 1 + 5.85iT - 83T^{2} \) |
| 89 | \( 1 - 7.59T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−9.276160270199888237534456934818, −8.699418540630842438060477390463, −7.67080775747604085817971507534, −7.00695707461713642215786326793, −6.22680624812043041519145717589, −5.03066632981753228759468918816, −4.60397814764430294895629274875, −3.75930312267292580042435349698, −2.53707084663430266790204403500, −1.58548260360480044960548834394,
0.42869395937112494562553423246, 1.55567784678942424456703100997, 2.93345015887056617409291653498, 3.44634228508358987142668241070, 4.91448855116982814611490616747, 5.48208830856537647071090102939, 6.41017783279591127998206783787, 7.08214822960570812415987998762, 7.936839977845112141929854188284, 8.711636286456799186165110104288